I have saved the proof of my job searches for Unemployment indefinitely since they don't take up much space on my computer. If you have a printer/copier/scanner just scan any paper copies of your job searches that you provide to your unemployment office and then it won't be necessary to keep paper copies taking up space that often gets lost. They'll be in a file on your computer. Be sure to back up your computer regularly, so that if nothing else, you'll have copies of everything that's on your computer on a flash drive or disc ready to load onto your new computer when you get one.
As far as keeping paper copies, I would think five years should be plenty of time. Next time you talk with one of the agents at the unemployment office, ask him/her this question. I doubt they check back more than a year or so at most unless they suspect fraud. I find keeping all my employment records in a file on my computer solves the problem.