So I just re-purchased the game, used, and I popped it in. It says "reading disc", then it says "playing DVD" shortly before fading into a white screen that says something like "To play this DVD please insert it into an Xbox 360" with something like seven variations of the same message in different languages.

The disc is mildly scratched, but not enough to affect playback in my experience (by which I mean that I've seen several instances of other games being scratched up worse and still play perfectly fine).

I tried applying water to the disc and using a fine cloth to gently clean it, but nothing. Anyone had this problem, and if so, how did you rectify it?

My 360 started doing that about half a year before it finally died.  It got to be more like an NES than an xbox-with having to re-insert the game over and over before it would finally work.  It might be on it's last legs.

Got enough space for a full HDD install?  Although if it's a disc reading problem, I wonder if you can even do that....

That's be unfortunate, since I've had the unit less than a year.