I do keep a paper list of important phone numbers in my emergency bag. However, I’ve been thinking about this problem and I think it makes sense to actually memorize some of the important numbers and actually make some calls using the keypad instead of just clicking on the name in my contact list.
A 3x5 card of the really important numbers stays in my wallet, in case I need to go old skool.
Phones do break at the most inopportune time.