I’d love to get a MAME arcade cabinet but they would just take up too much space, so I thought of building one into your average door. It would need wheels to support the weight of the flat TV/electronics, and a power source would be tricky to execute. All said, I think this is do-able.
Dave's Geeky Ideas 
Definitely doable! Raspberry pi or even a zero jammed in the door. Door never leaves the hinge so running power wouldn’t be an issue either.
Why wouldn’t you just put it on the wall? A narrow case no deeper than the TV mount ought to be able to hide wires and hardware and allow the playing panel to fold up when not in use, and could also look more like an arcade cabinet. Much less chance of damage as well.
A door could be swapped in and out. If you’re a renter this is handy, if you’re a homeowner then one built into a wall is more likely, but if you own a home, you probably have the space for a full-size cabinet. The doorcade is a space-saving design first and foremost.
Maybe I’ve just had a lot of bad luck with doors and good luck with landlords.