Astyanax Say the servers get to 100F and you need to cool them and you’re on the “night” side how do you get the -100 F vacuum of space in to cool it down? As you mentioned there’s no air or atmosphere to conduct the heat away from the servers.
Moving the heat isn’t the problem, it’s where to put it. They could pump it (like hot PCs do with liquid cooling) into the ground (if it’s cooler) or into water/ice (if they have enough). Otherwise it’s infrared radiators like the ISS uses (but that’s really inefficient). When it’s night it’s relatively easy because the sun isn’t blasting every visible surface with infrared and other radiation so everything cools off and water will freeze.
You just need SOMETHING that can absorb the heat from the computer and shed that heat through conduction or radiation at night faster than the data center produces it. Maybe a large field of water, ammonia, or some other fluid filled cells insulated from the sun light would do it. It has to be able to be modular and grow to keep up with the heat output of the data center.