To illustrate the point with a concrete example. You can heat something with the thermal transfer rate of aerogel to an absurdly high temperature and it will still be safe to pick up with your bare hand. Physics says it has a temperature but our intuition says something is wrong with the physics.
Yes I'm making that argument. Because it's true. The temperature of what particles do exist, 500km above the earth, is more likely to be in the thousands of degrees farenheit than below zero farenheit.
The discussion being had, if you read comments above your original, is that it's widely thought that "space is cold" and therefore it's good for cooling.
You're right that heat capacity means that the temperature of space is not relevant to its ability to cool a datacenter. You're wrong that making that argument is a good way to get people to actually change their mind.
Instead, attack the idea at its foundation. Space is not cold, not in the places where the data centers would be. It's much easier to get someone to understand "the temperature at 500km where the auroras are is very hot" than "blah blah heat capacity".
Now you see the point!
Budget the solar panel area as a function of the maximum computational load.
The rest of the satellite must be within the shade of the solar panel, so it basically only sees cold space, so we need a convex body shape, to insure that every surface of the satellite (ignoring the solar panels) is radiatively cooling over its full hemisphere.
So pretend the sun is "below", the solar panels are facing down, then select an extra point above the solar panel base to form a pyramid. The area of the slanted top sides of the pyramid are the cooling surfaces, no matter how close or far above the solar panels we place this apex point, the sides will never see the sun because they are shielded by the solar panel base. Given a target operating temperature, each unit surface area (emissivity 1) will radiate at a specific rate, and we can choose the total cooling rate by making the pyramid arbitrarily long and sharp, thus increasing the cooling area. We can set the satellite temperature to be arbitrarily low.
Forget the armchair "autodidact" computer nerds for a minute
So no, you cannot increase too much the height of the pyramid, there will be some optimum value at which the pyramid will certainly not be sharp. The optimum height will depend on how much of the pyramid is solid and which is the heat conductance of the material. Circulating liquid through the pyramid will also have limited benefits, as the power required for that will generate additional heat that must be dissipated.
A practical radiation panel will be covered with cones or some other such shapes in order to increase its radiating surface, but the ratio in which the surface can be increased in comparison with a flat panel is limited.
its CPU/GPU clusters, so we don't have 0 control on where to locate what heat generators, but even if we had 0 control over it, the shape and height of the pyramid does not preclude heat pipes (not solid bars of metal, but having a hot side where latent heat of a gas condensing to a liquid on the cold side and then evaporating on the hot side).
heat pipes have enormous thermal conductivities
Problem is with solar panels themselves. When you get 1.3kW of energy per square meter and use 325w of that for electricity (25% efficiency) that means you have to get rid of almost 1kW of energy for each meter of your panel. You can do it radiatively with back surface of panels, but your panels might reach equilibrium at over 120°C, which means they stop actually producing energy. If you want to do it purely radiatively, you would need to increase temperature of some surface pointing away from sun to much more than 120°C and pump heat from your panels with some heatpump.
Nevertheless, the problem described by you remains, the panels must dissipate an amount of heat at least equal with the amount of useful power that is generated. Therefore they cannot have other heat radiators on their backside, except those for their own heat.
Temperature: NaN °C
The universe is filled with such a bath of radiation, so it makes sense to say the temperature of space is the temperature of this bath. Of course, in galaxies, or even more so near stars, there's additional radiation that is not in thermal equilibrium.
I've always enjoyed thinking about this. Temperature is a characteristic of matter. There is vanishingly little matter in space. Due to that, one could perhaps say that space, in a way of looking at it, has no temperature. This helps give some insight into what you mention of the difficulties in dealing with heat in space - radiative cooling is all you get.
I once read that, while the image we have in our mind of being ejected out of an airlock from a space station in orbit around Earth results in instant ice-cube, the reality is that, due to our distance from the sun, that situation - ignoring the lack of oxygen etc that would kill you - is such that we would in fact die from heat exhaustion: our bodies would be unable to radiate enough heat vs what we would receive from the sun.
In contrast, were one to experience the same unceremonious orbital defenestration around Mars, the distance from the sun is sufficient that we would die from hypothermia (ceteris paribus, of course).