What is the best way to live off of a set amount e.g. $100k - $150k in savings with no income? Assuming you can relocate to anywhere in the US with low cost of living, can you live off of this for 20+ years?
The terms you have set are on the edge of possibility with luck and much sacrifice, but definitely not recommended. You will have a meager existence at best.
The MIT-estimated poverty wage (not living wage, that's hopeless) for a single adult bottoms out in states like SD, AR, and KY at $5.84/hr or about $12k/yr. If you stretch to the upper savings amount of $150k, then to last (barely) 20 years, you would have to earn 5%/yr above inflation (whatever inflation rate you need to grow the withdrawals). This is an aggressive return target; you would have to invest heavily in stocks and take considerable risk.
There is no way to achieve your goal easily. To live decently (living wage for a single adult, still no luxuries) with some certainty for 20 years, you would need at least $300k.
Head for Alaska.
Assuming you have access to some kind of training and common sense, you could purchase a small plot of land and build a small cabin for very little out there and live off the land.
Even if you allocate up to $35k for the initial purchase, move, and supplies, you could conceivably establish a homestead and operate off survival skills, hunting, trade or barter, and any other non-cost based need for the region. The remaining would be kept in an account and separated into low risk certificates so they do earn money, albeit not a great deal.
The remaining $65k (give or take) would only be used for non-renewable resources like ammunition or specialized items or repairs, like a furnace or other appliances you aren't able to acquire through trade. You would also need to know the tax laws regarding your land. Factor that in expected annual expenses and make sure your certificates or investments cover the property taxes.
It is conceivable a well skilled outdoors person could live their entire lives under these conditions if the proper trade system was in place. The land provides all your body needs to live and it is possible to hunt without ammunition. Your actual costs for living would only be within the constructs of the human architecture. Like medical expenses, for example. But if you were willing to take the risk that you could navigate the health issue, and mandates like taxes and such, you would be limited only by your own ability to survive in a harsh environment.
It's probably unnecessary to say this, but don't do this unless you actually know what you're doing. Watching a few seasons of one of those Alaska people shows won't prepare you for anything you'll face out there. Plus, it might cost you the entirety of your savings to get you out of that place if things go wrong. I'm just posting this to say it is possible.
You can probably find 3 bed/2 bath homes for rent in a decent suburb in a cheap area for $1500/mo. Such a home should be able to sleep 6 such people. $1500/6 = $250/mo. Figure $300/mo utilities and it's $300/mo. That leaves $116/mo for food and clothing. You need 45,000 cal per month. You can buy rice in bulk for $0.40/lb, and a pound of rice is 435 calories... so about $45/mo (throw some beans in instead of some of that rice, they're similar or better in $/cal). You've now got $71/mo for clothes and soap, should be easily doable. For entertainment, talk to your roommates or go for a walk.
What is the best way to live off of a set amount e.g. $100k - $150k in savings with no income?
There are enough assistance programs that many people ill-prepared for retirement get by. There's help with rent, food, healthcare, property taxes, etc. If you qualify for enough of those programs then you can live off very little savings.
If you don't qualify for any/enough assistance then you'd have to be thinking about an alternative lifestyle. Off-grid homesteading. Small plot of land, build a tiny cabin with no electricity, dig a root cellar/larder, grow/raise your own food. You can live off a small amount of land pretty much indefinitely. Use your money for simple tools.
Maybe you earn non-wage income by bartering services for room/board, I know people who lived rent-free with senior citizens in exchange for running errands and helping around the house.
The best way to live with little money will vary wildly from person to person, but without assistance $150k is likely not enough to live 20+ years on your own without a lot of luck or hard work.