I think it would work. We have our media junk (Tivo, router, etc.) in a cabinet with solid walnut doors. It has a lot of space and a small ventilation grid in the back (which you won't be able to do, so an attractive front treatment is in order). Louvered doors might be an option, depending on the room style. Or cane...and of course, that old woven speaker covering stuff if you feel MCMish. The pictured mesh is a little country for me, but would be lovely in the right place.

I don't know about Xbox, but other remotes will be fine with the door shut.

If any of the devices inside the cabinet needs to receive infrared signals from a remote, as a cable box does, a grill like that may interfere or block the signals and thus not work. Often seen for such front panels, for this very reason, is tinted glass.

For ventilation, I think if you had some good sized holes drilled in the sides (toward adjacent compartments) and back of the compartment, several holes of a few inches in diameter, and put bumper pads on the front doors so that they don't close all the way, or have them mounted and aligned so that there's a good but imperceptible gap around the edges when they close, you should be fine. If you're concerned, you could always mount a small exhaust fan blowing to the side or back and maybe have it connected to come on with the cable box.

If you find a remote signal can pass through a mesh, it should be more than enough to prevent too much heat build up. In my built-in in a section with one door, put a couple of bumpers on top of each other to make a pretty good size gap, and also drilled a 2 inch hole to the side, and it keep it from getting too hot. I've since switched to a Roku dongle and ditched the cable box so that the problem is gone but for a DVD player used only occasionally.