There used to be some sort of implied trust between banks where you could go to any atm machine and withdraw some cash even if the communications were down between the banks. Then if/when the communication was restored and you didn't have the funds you got an overdraft fee from your bank. I don't know if this has changed or not with all the new laws.
Best thing to do is keep some cash on hand for emergencies and then use direct deposit and online banking for the rest. Then if you do have to evac you have some spending cash for whatever needs you have and your company can still get a paycheck to your account which you could still access online somewhere and pay a few bills.