Is it possible to read every book in Skyrim?
Yes
An easy way to accomplish this if you are on PC is to use a console command to take you to a room that contains every single item in the game, including a chest with all of the books in the game.
Use the following command to get to the room:
- coc qasmoke
Use this command to exit the room:
- coc whiterun
If you are on console you will have to do @Nelson method
Yes.
It's relatively simple. If your goal is to acquire all the books, there are book vendors that you can buy almost everything from, then it is a matter of using online resources to find the ones you are missing.