I wouldn't put a Pi in it either.
This dude has some cheapy computers for sale if your looking:
http://forum.arcadecontrols.com/index.php/topic,133968.0.htmlOr you could always hunt up an HP Slimline or similar pretty cheap on eBay.
If your trying to minimize space, buy a mini ITX board and small form factor power supply. Pair it up with a low wattage CPU. It's MAME, no need for an 8 core 125 watt CPU.
For graphics a GT 730 makes an excellent choice in price, performance, and dead silence since it uses no fan. Or, if you don't want a card sticking out, any of the AMD AM1 series boards with integrated graphics works well. Cheap too.
So many ways to go here... not really a matter of what to get, but rather how much do you want to spend?