Here is some more analysis on the problem of radiating heat away in space: https://toughsf.blogspot.com/2017/07/all-radiators.html
Giant radiators:
http://claudelafleur.qc.ca/images/iss028e005676.jpg
https://en.wikipedia.org/wiki/External_Active_Thermal_Contro...
Fluid pipes are threaded through them for transferring heat. These contain mixed-phase ammonia: I believe it should condense from vapor to liquid under the radiators, the coldest point. That phase change adds a huge boost to their heat-carrying capacity.
You can radiate heat from the surface of the earth into space via the sky, but you don't lose much energy that way.
That's not quite true. Black body radiation is the only reason why earth does not heat up much more. It's the only factor with a negative radiative forcing.
Solar radiation is the only main incoming source of energy, and black body radiation the only outcome with almost the same magnitude (which is huuuuge btw)
The actual surface of the earth has the advantage of being able to use the whole surface of the earth.
On earth you can use things like fans and evaporative cooling. It's far superior to making huge space radiators, but nothing is free.
In space there is mostly no atmosphere, you have to radiate that heat which isn't quite as easy as it sounds: https://www.space.com/21059-space-station-cooling-system-exp...
An object in space loses heat according to the Stefan-Boltzmann law of radiation. That use the fourth power of the temperature differential so you get boost from the background of space being -270C. BUT the Stefan-Boltzmann constant is 5.67 × 10−8 J/s · m2 · K4 and that is the problem.
You can do the numbers yourself but various internet sources suggest that spacecraft radiators can only cool between 100 and 350 W of internally generated heat per square meter. That is a big radiator for not much cooling.