1. Hard copies - I purchased a bill organizer for $2 from Dollar Tree, or you can use an old 3-ring binder you have laying around. Print out montly bill paid receipts from your online accounts, hole punch them, and store them in the binder by month (not category). This way, you have a year-at-a-glace of bills paid in case the question arises.
2. Calendars - I trust my iPad calendar as much as the next gal, but also keep a tiny 3-ring binder calendar by Greenroomeco.com that I found at Target. You can buy refills and continue to use the cute binder for years to come. I jot down appointments, things I need to do, a shopping list, and many other things in my calendar.
3. Contacts - Again, I trust my iPad to hold my contacts. I can access the contacts even without an internet connection. Still, I have a cheapo $1 phone book that I found at Dollar Tree that has emergency contacts and some friends. You never know when you will need that. At minimum, you can leave it with the babysitter. You wouldn't want her/him having your iPad.
4. VItal documents - You should invest in a small fire-proof safe with a handle (they are heavy) and use it to store your birth certificates, marriage license, social security cards, passports, deeds, and other vital documents which would be a major pain to replace. I love my little safe! I even store special photos that I could not do without (before I had a digital camera!).
Let me know what you do in this kind of situation. What are your back-ups?