(no noise, no moving parts, no heat).
Really? That's easier said than done.
For the CPU you can use one of the VIA processors with the Nehemiah core. They should run the old classic games well enough, but I don't know for sure. You can run caseless so that more air can get to the CPU and video card.
There shouldn't be a problem with the video card. There are plenty of passively cooled cards out there.
There are some passively cooled power supplies, too, such as the ProSilence PCS-350. They may be expensive, however.
Is convection alone going to be enough to exhaust all the heat from your components? Don't forget exhaust vents at the top of the case.
This is very interesting, I hope you are successful in creating a cab with no moving parts on the inside.
Buddabing