By my back of the envelope calculations, the radiators would be comparable to the solar arrays, probably somewhat smaller and not massively bigger at least.
Extremely rough one significant digit analysis from first principles, containing a lot of assumptions:
For solar panels:
Assuming area of 1000 square meters (30m x 30m square), solar irradiance of 1 kW/m^2, efficiency of 0.2. As a result power is 200 kW.
For radiators:
Stefan-Boltzmann constant 6E-8, temperature difference of 300 K, emissivity of one, we get total radiator power 1000 x 6E-8 x 300^4 = 486 kW.
The radiator number is bigger so the radiator could be smaller than the solar panels and could still radiate away all the heat. With caveats.
Temperature difference in the radiator is the biggest open question, and the design is very sensitive to that. Say if your chips run at 70 C (340 K), what is the cool temperature needed to cool down to, what is the assumed solar and earth flux hitting the radiator, depends on geometry and so on.
And then in reality part of the radiator is cooler and radiates way less, so most of the energy is radiated from the hot part. How low do you need to get the cool end temperature to, in order to not fry your chips? I guess you could run at very high flow rates and small temperature deltas to minimize radiator size but then rest of the system becomes heavier.
There's a very clever scheme I remember reading about a while ago where you dump the heat into an oil that you then spray in a fine mist towards a collector. You get a collosal surface area that way, in a very confined volume, with not that much more mass than a coolant fluid which you already need; and it's relatively easy to homogenise the temperature across the radiating particles. I seem to recall that it got as far as Dupont coming up with a specific coolant mix for the job; the rest of the system is a relatively well-understood (if precise) nozzle/collector design so you don't end up squirting your coolant off somewhere you can't catch it.
In space this wouldn't really work since there's no conduction or convection.
If you think of a big ball of droplet mist. From the point of view of a droplet in the center, it gets heat radiation from all the droplets around it. It can only radiate heat to black sky it sees, and it might be none, it's "sky" is just filled by other hot droplets. So it doesn't cool at all.
The total power radiated can't exceed the proportion to the macro surface area with tricks.
Question: for a larger system, can a heat pump be used to increase the temperature of the radiator without making the rest of the system hotter? Thus radiating more heat from fewer panels?
Your temperature differential is already 300K, so the efficiency needs to be high enough. 50K change is only 18% more cooling, but if COP=5 then it's also putting out 20% more heat...
In addition to the math, you can also look at existing examples, like how large the ISS radiators need to be relative to its solar panels. Like this project, it is essentially a closed system where all power generated by the solar panels will eventually be converted to heat that needs to be dissipated.
I'm skeptical that it makes any economic sense to put a datacenter in orbit, but the focus on the radiators in the last discussion was odd - if you can make the power generation work, you can make the heat dissipation work.
