Ask Dr. Glowcat: Can We Has Data Centres in Space?
Sure. The same way you can have a football field on top of mount Everest.
Oh, you meant in your science-fiction novel? Then yes, absolutely, but you would also have to come up with some meowgic technology for cooling, because there's no cooling in space. Vacuum does not conduce heat, so things in space cannot disperse the heat passively.
Then there's the massive expense of replacing the GPUs every... uh, few days, probably? The cosmic rays and solar winds would fook them up real quick, and cause all sorts of bugs in the meantime. The hardware that goes in satellites and space-borne scientific instrument is specifically designed for going to space, because it has to go to space. The most hostile environment that exists. You see, I've actually worked with satellites for astrophysics research, so I've seen a thing or two up-close1.
But in a science fiction book you can just brush everything under the carpet, or invent all sorts of plausible-sounding technomancy to fix the cooling. Or it could be a plot point, when the data centre cooling thingymajig breaks down and all the robots in the galaxy stop working. There, you have your free novel idea for the week! And I'll go back to procrastinating the work I actually need to do.
well, figuratively, as both satellites were already in space when I started working with them.↩