Physical Danger
Are you kidding? Have you ever gotten hurt in the library? Sure, Timmy did that one time, but he's learned not to climb on the bookshelves any more. Simply put, the biggest threat to life and limb is probably going to be paper cuts (the horror). If you work with electronic media, even that isn't very likely.
There's always the chance that your line of work will involve some kinds of chemicals to help preserve older documents. Those can be dangerous if mishandled, but if you've ever worked in a photo lab you'll probably be okay. If you don't know what a photo lab is, go find an old person and ask them.