I stumbled onto a solution to the error message that pops up and tells you the disc can't be read and to take it out and clean it.

Go to the xbox main screen and page over to the menu that lets you look and settings and storage. Go to storage and find you list of saved games. One by one try to move the saves to a new storage device. You don't actually have to move anything...if the game asks you if you want to move the file, just say no and try the next save. What you will eventually will find is a saved game file that will give you an error message telling you this file can't be moved there is a problem. Delete this saved game. Go back to the main start up screen a try restarting the game and it should it should be fine. it seems that if a save file gets corrupted it treats it like the disc is bad.
This has happened to me twice and both times I fixed it using the above method.

Are you sure it's not the actual disc that's kind of broken or just scratched?

Positive it is not the disc. It never had any scratches, but went ahead and actually took my disc to a video store and paid to have it thoroughly cleaned. And that had no effect on the error. Only after a month later, and starting over with a new character did I find the fix listed in the above post.