My banking and investment records will be up to date, in case a computer scrambles some data. In December, I'll get printouts of the payments I've made on mortgages and other loans. I might even get a copy of my credit report. Bank accounts and securities. My money stays put, where it's FDIC-insured.
Securities stay put, too.
Don't worry about the ATMs. If your usual automated teller machine fails, there are others around. Banks will have extra staff on call. This is one of the things your bank should be giving you information about. Find out if it has tested its link to the Social Security Administration, the Department of Veterans Affairs, your employer or any other source that sends payments directly to your bank account.