Backup disk suddenly won't work
I've had this problem before. I think it is because the data is corrupt. What you need to do is restore the drive and create a new one. The easy way is to get a different computer and reformat it there, but if not, don't worry. Just go into Disk Utility and attempt to restore/reformat the drive. If it gives you an error like "Can't unmount the disk", follow this guide to forcefully unmounting a Time Machine backup disk. Then, you will have to setup a new backup.

Note: If your drive still gives problems after this, it may be broken and you may need to purchase a new one.