The first step to placing the Xbox 360 into the Atari case is to disassemble each for their needed parts. The Atari would need to have its guts removed and the Xbox 360 would have to have its motherboard removed (along with any other needed parts).
Taking apart the Atari is as simple as removing four screws on the bottom of the console. The Xbox 360, on the other hand, was designed to be difficult to take apart. Fortunately, we disassembled the Xbox 360 and wrote up a detailed description in a previous article. Click here for detailed instructions and a video illustration the five-minute teardown. Figure 1 and Figure 2 provide a nice compare/contrast of the Atari circuit board vs. the Xbox 360.
Figure 1: Atari circuit board
Figure 2: Xbox 360 motherboard