Cooling via radiation is relatively slow compared to convection, but remember, your background is essentially -270 degrees. Radiative cooling scales with the temperature difference to the fourth, so it’s not as weak as you may think.
When cooling *does* become an issue, large black radiator panels are an option. These radiate heat away quickly, so if heat can be transported to these panels it radiates away much quicker than it would from the craft body.
It’s really not that slow.
Space is very cold (usually) and radiative cooling scales to the fourth power. Which means that if you double the temperature difference an object will cool 16 times faster.
Because of that cooling stuff in space is (very very roughly) akin to trying to cooling stuff in a giant refrigerator. As far as how quickly the background will try to suck away heat.
So sure, if something is producing a lot of heat that’s a problem. But then you just increase It’s surface area and you are fine again.
Space stations have big radiators, they’re big panels that are ideally kept edge on to the sunlight, and the hot coolant is circulated through them to radiate the heat out into space. Though even if it’s in sunlight, you can still get some heat rejection if your coating is highly reflective to visible light, but not infrared: https://www.youtube.com/watch?v=dNs_kNilSjk
Space suits let some water boil off/sublimate into space to dump heat. https://youtu.be/6mMK6iSZsAs?t=1942
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