That's all well and good but not particularly helpful. Everyone has their opinions about what the best way to backup is, but since this is a technical forum we try to help people with their particular problems rather than offering alternative methods.
His problem is a specific and technical one, a problem that affects other things besides specifically backing up his Xbox games. I'm sure he'd like the problem solved, not recommendations regarding a new hard drive.
If you're having problems ripping mp3s to your computer, do you want someone in a tech forum telling you to just make CD-R copies instead?