In general, you shouldn't think of space as being cold for intuitive hot-to-cold heat transfer. For example, the metallic side of a space ship would not be anywhere near 0K, whereas if you had a metal plaque between a liquid at around 0K and your hand, it would be.

It is very hard to dissipate heat from a solid object into space. This is not true for our bodies on the other hand, but that is more to do with pressure - if you expose cells to the Void of space, most liquids inside would quickly expand in size and essentially boil, consuming large amounts of heat to go through the phase transition from liquid to gas, thus quickly cooling surrounding tissue. You could theoretically use this to create evaporation-based heating, but you would have to transport vast quantities of water that would quickly be used up, since there is no hope of collecting them back most likely.

Shedding heat in space is a huge problem because you can only radiate it. Depending on the refrigeration requirements there may be too much excess heat to make it feasible.

You’d probably lose a lot of the advantages because you lose the naturally extremely effective radiation shielding of the earths magnetic field and atmosphere.

You'd probably always have to stay in the shadow of some celestial body and the 3K microwave background would be an issue as well that would have to be solved.