>vacuum is a fucking terrible heat convector
Yes we're talking about radiating not convection
And a kilowatt from one square meter is awful. You can do far more than that with access to an atmosphere, never mind water.
Assuming that this is the right order of magnitude, a 8MW datacenter discussed upthread would require ~8000 m^2, plus a fancy way of getting the heat there.
A kilowatt is nothing. The workstation on my desk can sustain 1 kW.