No, not that Doom... the other one... you know, the first one. Anyways, some nutty XBox hacking project called OpenXDK has posted a new version of their XBox development kit with the ability to compile programs(!), and:
To demonstrate the capabilities of the current OpenXDK framework, the SDL port of Doom (without sound or networking) is able to be compiled and executed. You will need to find a DOOM1.WAD (shareware) and put it in the root of your XBOX's C drive.
So uh, it works, sort of. Hey couldn't you just get Linux running and play Doom that way? Whatever, that's probably not the point.