I am looking for a file hosting service that allows for FTP so that I can backup my files. Any recommendations?

Are those files on your PC or you want to backup your hosting account.

For servers, bqbackup.com, rsyncpalace.com, webbycart.com are quite popular and they are good.

Also visit http://www.webhostingtalk.com/forumdisplay.php?f=132 there are some good offers there.

How much disk are you backing up? The size can determine which service is best for you.