I'm afraid you have missed a few of the outcomes commonly faced by millions of Americans, so I would like to take a moment to discuss a wider range of outcomes that are common in the United States today. Most importantly, some of these happen before retirement is ever reached, and have grave consequences - yet are often very closely linked to financial health and savings.
Not planning ahead long-term - 10-20+ years - is generally associated with not planning ahead even for the next few months, so I'll start there.
Unexpected Life Event Risk
The most common thing that happens is the loss of a job, or illness/injury that put someone out of work. 6 in 10 adults in the US have less than $500 in savings, so desperation can set in very quickly, as the very next paycheck will be short or missing. Many of these Americans have no other source of saved money, either, so it's not like they can draw on retirement savings, as they don't have that either. Even if they are able to get another job or recover enough to get back to work in a few weeks, this can set off a desperate cycle.
Those who have lost their jobs to technical obsolescence, major economic downturns, or large economic changes are often more severely affected. People once making excellent, middle-class (or above) wages with full benefits find they cannot find work that pays even vaguely similarly. In the past this was especially common in heavy labor jobs like manufacturing, meat-packing, and so on, but more recently this has happened in financial sectors and real estate/construction during the 2008 economic events. The more resilient people had padding, switched careers, and found other options - the less resilient, didn't.
Especially during the 1970s and 1980s, many people affected by large losses of earning potential became sufficiently desperate that they fell heavily (or lost their functioning status) into substance abuse, including alcohol and drugs (cocaine and heroine being especially popular in this segment of the population). Life disruption - made even more major by a lack of savings - is a key trigger to many people who are already at risk of issues like substance addiction, mental health, or any ongoing legal issues.
Another common issue is something more simple, like loss of transportation that threatens their ability to hold their job, and a lack of alternatives available through support networks, savings, family, and public transit. If their credit is bad, or their income is new, they may find even disreputable companies turn them away, or even worse - the most disreputable companies welcome them in with high interest and hair-trigger repossession policies.
The most common cycle of desperation I have seen usually starts with banking over-drafts, and its associated fees. People who are afraid and desperate start to make increasingly desperate, short-sighted choices, as tunnel-vision sets in and they are unable to consider longer-term strategy as they focus on holding on to what they have and survival.
Many industries have found this set of people quite profitable, including high-interest "check cashing", payday loans, and title loans (aka legal loan sharks), and it is not rare that desperate people are encouraged to get on increasing cycles of loan amounts and fees that worsen their financial situation in exchange for short-term relief. As fees, penalties, and interest add up, they lose more and more of their already strained income to stay afloat.
Banks that are otherwise reputable and fair may soon blacklist them and turn them away, and suddenly only the least reputable and most predatory places offer to help at all - usually with a big smile at first, and almost always with awful strings attached.
Drugs and alcohol are often readily available nearby and their use can easily turn from recreational to addictive given the allure of the escapism it offers, especially for those made vulnerable by increasing stress, desperation, loss of hope, isolation, and fear.
Those who have not been within the system of poverty and desperation often do not see just how many people actively work to encourage bad decision making, with big budgets, charm, charisma, and talent. The voices of reason, trying to act as beacons to call people to take care of themselves and their future, are all too easily drowned out in the roar of a smooth and enticing operation.
I personally think this is one of the greatest contributions of the movement to build personal financial health and awareness, as so many great people find ever more effective ways of pointing out the myriad ways people try to bleed your money out of you with no real concern for your welfare. Looking out for your own well-being and not being taking in by the wide array of cons and bad deals is all too often fighting against a strong societal current - as I'm sure most of our regular contributors are all too aware!
Jail and Prison Risk
With increasing desperation often comes illegal maneuvers, often quite petty in nature. Those with substance abuse issues often start reselling drugs to others to try to cover lost income or "get ahead", with often debilitating results on long-term earning potential if they get caught (which can include cost barriers to higher education, even if they do turn their life around).
I think most people are surprised by how little and petty things can quickly cycle out of control. This can include things like not paying minor parking or traffic tickets, which can snowball from the $10-70 range into thousands of dollars (due to non-payment often escalating and adding additional penalties, triggering traffic stops for no other reason, etc.), arrest, and more.
The elderly are not exempt from this system, and many of America's elderly spend their latter years in prison. While not all are tied to financial desperation as I've outlined above, a deeper look at poverty, crime, and the elderly will be deeply disturbing. Some of these people enter the system while young, but some only later in life.
Rather than homelessness being something that only happens after people hit retirement, it often comes considerably earlier than that. If this occurs, the outcome is generally quite a bit more extreme than living off social security - some just die.
The average life expectancy of adults who are living on the street is only about 64 years of age - only 2 years into early retirement age, and before full retirement age (which could of course be increased in the next 10-20 years, even if life expectancy and health of those without savings don't improve). Most have extremely restricted access to healthcare (often being emergency only), and have no comforts of home to rest and recuperate when they become ill or injured.
There are many people dedicated to helping, yet the help is far less than the problem generally, and being able to take advantage of most of the help (scheduling where to go for food, who to talk to about other services, etc) heavily depends on the person not already suffering from conditions that limit their ability to care for themselves (mental conditions, mobility impairments, etc).
There is also a shockingly higher risk of physical assault, injury, and death, depending on where the person goes - but it is far higher in almost every case, regardless.
Never Making It To Retirement
One of the chief problems in considering only retirement savings, is it assumes that you'll only have need for the savings and good financial health once you reach approximately the age of 62 (if it is not raised before you get there, which it has been multiple times to-date). As noted above, if homelessness occurs and becomes longstanding before that, the result is generally shortened lifespan and premature death.
The other major issue of health is that preventative care - from simple dentistry to basic self-care, adequate sleep and rest, a safe place to rejuvenate - is often sacrificed in the scrambling to survive and limited budget. Those who develop chronic conditions which need regular care are more severely affected. Diabetic and injury-related limb loss, as one example, are far more likely for those without regular support resources - homeless, destitute, or otherwise.
Limited Post-Retirement Benefits
Other posters have done a great job in pointing out a number of the lesser-known governmental programs, so I won't list them again.
I only note the important proviso that this may be quite a bit less in total than you think. Social Security on average pays retired workers $1300 a month. It was designed to avoid an all-too-common occurrence of simple starvation, rampant homelessness, and abject poverty among a large number of elderly. No guarantee is made that you won't have to leave your home, move away from your friends and family if you live in an expensive part of the country, etc. Some people get a bit more, some people get quite a bit less. And the loss of family and friend networks - especially to such at-risk groups - can be incredibly damaging.
Note also that those financially desperate will be generally pushed to take retirement at the minimum age, even though benefits would be larger and more livable if they delayed their retirement. This is an additional cost of not having other sources of savings, which is not considered by many.
Well, yes, many cannot retire whether they want to or not.
I cannot find statistics on this specifically, but many are indeed just unable to financially retire without considerable loss. Social Security and other government plans help avoid the most desperate scenarios, but so many aspects of aging is not covered by insurance or affordable on the limited income that aging can be a cruel and lonely process for those with no other financial means. Those with no savings are not likely to be able to afford to regularly visit children and grandchildren, give gifts on holidays, go on cruises, enjoy the best assistive care, or afford new technological devices to assist their aging (especially those too new and experimental to be covered by the insurance plans they have).
What's worse - but most people do not plan for either - is that diminished mental and physical capacity can render many people unable to navigate the system successfully. As we've seen here, many questions are from adult children trying to help their elderly parents in retirement, and include aging parents who do not understand their own access to social security, medicaid/medicare, assistive resources, or community help organizations.
What happens to those aging without children or younger friend networks to step in and help? Well, we don't really have a replacement for that. I am not aware of any research that quantifies just how many in the US don't take advantage of the resources they are fully qualified to make use of and enjoy, due to a lack of education, social issues (feeling embarrassed and afraid), or inability to organize and communicate effectively. A resource being available is not very much help for those who don't have enough supportive resources to make use of it - which is very hard to effectively plan for, yet is exceedingly common.
Limited Aging and End of Life Support
Without one's own independent resources, the natural aging and end of life process can be especially harsh. Elderly who are economically and food insecure experience far heightened incidence of depression, asthma, heart attack, and heart failure, and a host of other maladies. They are at greater risk for elder-abuse, accidental death, life-quality threatening conditions developing or worsening, and more.
Too Grim a Picture?
Scare-tactics aren't always persuasive, and they do little to improve the lives of many because the people who need to know it most generally just don't believe it. But my hope here is that the rather highly educated and sophisticated audience here will see a little more of the harsher world that their own good decisions, good fortune, culture, and position in society shields them from experiencing. There is a downside to good outcomes, which is that it can cause us to be blind to just how extremely different is the experience of others.
Not all experience such terrible outcomes - but many hundreds of thousands in the US alone - do, and sometimes worse. It is not helpful to be unrealistic about this: life is not inherently kind.
However, none of this suggests that being co-dependent or giving up your own financial well-being is necessary or advised to help others. Share your budgeting strategies, your plans for the future, your gentle concerns, and give of your time and resources as generously as you can - within your own set budgets and ensuring your own financial well-being. And most of all - do not so easily give up on your family and friends, and count them as life-long hopeless ne'er-do-wells. Let's all strive to be good, kind, honest, and offer non-judgmental support and advice to the best of our ability to the people we care about. It is ultimately their choice - restricted by their own experiences and abilities - but need not be fate. People regularly disappoint, but sometimes they surprise and delight. Take care of yourself, and give others the best chance you can, too.