If you play your games on Steam you could use their in home-streaming service, put your computer in another room and stream to a barebones PC in your box room.

I believe Steam lets you stream non Steam games too.

If you play your games on Steam you could use their in home-streaming service Steam Link for ~50$, put your computer in another room and stream to your monitor in your box room.

Steam Link doesn't produce any warmth that matters, it's as small as a hard drvie and has inputs for 3 USB-devices, LAN and W-LAN. HDMI as output.

I believe Steam lets you stream non Steam games too.