Assuming one or more of the big investment banks incurs unsustainable investment losses leading to another banking crisis, what is the safest place to invest my 401k?
You have a few options, depending on what you believe would be safer. This also depends on when you would like to access your 401k. If your going to be retiring in 10 years then you wouldn't care what happens in the next 3 years so much.
I will assume your accessing your retirement account already...
If you are scared of a banking collapse you can always just hold your money in cash, it is the most liquid. Unless you have more then what the FDIC will insure that is probably your safest bet.
You have the option of putting your money in bonds. Historically, treasuries are the safest.
TIPS - Is another vehicle you can use to protect your self against inflation as well.
If your not a fan of treasuries then you can choose to invest in bonds of highly rated multinational corporations with a very healthy balance sheet, preferably companies with a huge cash pile, they would be the least influenced by a credit crunch. However even these companies would suffer unless investors deem governments unable to pay back their debt, which in that case they might flock to these bonds for safety.
Traditionally gold has also served as a safe haven and a hedge against inflation. It is essentially a currency that cannot be manipulated by the interest rate policies of any one government.
More advanced options (only for the experienced)
In a 401k you can't short sell directly or take a margin position so you are limited with this account. You are still able to hedge by going long on put options.
However with your personal account you can use more complicated trades. I won't go into heavy detail as they should only be reserved for more experienced investors. If your interested I suggest reading up on synthetic positions
In conclusion: You have a few options but regardless of what you choose there is some sort of risk in all, depending on what you believe. Now with that said I am not in the camp to believe a catastrophic financial collapse is just over the hill, so I am more inclined to say depending on your investment goals I would maintain a diversified portfolio in cash, bonds and gold.
Edit: I wanted to add, the only thing a banking crisis can guarantee is liquidity issues, so the safest asset would be the most liquid asset or what would be perceived as the most liquid. Inflation wouldn't be the result of a banking crisis directly, instead it could be the result of the actions from what The Fed might or might not do during a banking crisis.
I'm going to take a broader view on your question, on the assumption that you're NOT anywhere near retirement (the answer is totally different if you're 5 or even 10 years from retirement).
The safest thing you can do is ... drumroll ... have a diversified, balanced portfolio of stocks, bonds and cash instruments.
Not the answer you were looking for, huh? Look, like the others mentioned, savings accounts are FDIC insured. But, that's not the only risk you face. 401(k)s are for retirement, so they have to do two more things: 1) keep up with inflation, and 2) grow. A savings account, or any other "safe" investment, will do neither. In other words, if you keep it in a savings account, then in retirement dollar-for-buying-stuff-at-the-store dollar, YOU WILL GUARANTEE YOUR LOSSES.
Look back over the worst possible investment era's in the US: 1930-45, 1971-81, and 2000-now, and what do you find? A sensible portfolio of stocks and bonds, diversified and rebalanced, across any 40 years worth of saving for retirement, including those bad years, and you do ok -- even if you retire in the worst possible year.
Assuming you're in the US, the safest place to invest your 401(k) funds, bar none, is a bank account. (You would need to make sure that it's a 401(k) account and a FDIC insured bank account. I'm pretty sure these exist.) This account could be in the form of a savings account, or a Certificate of Deposit. (If you choose a CD, be aware of the restrictions on when you can redeem it without penality.)
It doesn't matter if the bank fails because you will be protected up to $250,000 by federal deposit insurance - the bank's shareholders will be stuck with a worthless bank, but you won't lose a dime. If you have more than $250,000, use two (or more) banks. Note that this safety comes at a cost. You will make very little interest on your savings. Its purchasing power may be eroded over time by inflation.
An alternative rather-safe place to store your money which will avoid inflation risk is Treasury Inflation-Protected Securities (TIPS). These are a type of bond issued by the US government which will pay you based on the Federal Reserve's Consumer Price Index, as well as a certain level of interest. Note that if you buy this type of bond, you will be subject to interest rate risk: in an inflation scenario, it is plausible that bond yields will increase, which means bonds' face value will decrease, and if you have to sell them before their maturity you may realize a loss. This will also be the case if you buy a TIPS mutual fund or exchange-traded fund. However, if you hold the bond to maturity, you will not realize any loss.
I don't think either of these are good ideas for most people, at least not for all your 401(k) portfolio, but it does make sense for some portion of your portfolio if you are approaching retirement or in retirement and you expect to spend the money that you place in either of those instruments (bonds or bank accounts) within the next several years.