Typically funds are available by the next business day, at most within a few days: http://www.snopes.com/business/bank/fdic.asp

No bank statements or formal claim for reimbursement is needed; the process is entirely automatic and usually done in less than 24 hours.

> If a bank failed, how long would it take to be reimbursed by the FDIC? Probably much longer than a poor person could weather.

In most cases, the FDIC takes over the bank and runs it (in a matter of hours), specifically to prevent this sort of impact.