Googling brought me to these pages. Fair warning, I just gave these sites a casual glance, so if there's any weird stuff on here, don't blame me...blame Google.
Installing emulators on an Xbox 360
Connecting the 360 to a NAS DLNA server (dunno if this'll work for MAME specifically, but it's a good first step).
I'm at work so I cant see any of those sites so I'll check at home. My guess is my 360 will probably need to be hacked.
Yup. It's like having to jailbreak an iPad to install 3rd party unofficial apps. They even mention it right up front, requiring something called a JTAG'd 360, which I assume is the name of the hack.