Space feels cold if you have some fluid to evaporate, like blood or something. But servers will very quickly run out of whatever fluids they have if they tried this. (And so would you in their place.)

The only option to sustainably lose heat in space is radiation, which works, but is slow and limited in capacity, so these server satellites would need massive radiators. It’s not impossible to do. ISS also has massive radiators.

So servers in space is possible. How big you can make an orbital server park, I don’t know. I can imagine that with enough radiators, they start catching each other’s heat, so there might be a limit to have many radiators you can put closely together, but I have no idea what that limit might be.