No different from the situation 500 years ago where a fire could destroy important records and leave countries in chaos. All banks need to do is keep track of how much money is in everyone's accounts and allow people to transfer money somehow. They do not need to be able to "give it back to people." Fiat money could exist on pen-and-paper ledgers too; in fact that was exactly what the earliest known money was (well, stylus-and-clay as neither pens nor paper had been invented yet).

If for some reason all our computers simultaneously failed without any warning, it would obviously be chaos, but at the end of the day the banks would be in a much better position to recover than anything like Bitcoin. Banks only use computer networks because they are more efficient than the alternatives; Bitcoin requires everyone to be online and does not have any alternatives. The worst thing that would happen if all computers failed simultaneously is that the banks would lose track of everyone's accounts -- a general amnesty for debtors and a big loss for people without debts -- and in the aftermath the banks would just start from scratch using older technologies.