James Webb, being an infrared telescope, has to be extremely cold: It's Mid-Infrared Instrument is actively cooled to 7°K (-266°C).
Most other instruments (NIRCam, NIRSpec, FGS/NIRISS) operate at about 40 Kelvin (-233°C) using only passive cooling (i.e. the sunshield).
Clearly this would be cold enough for a data center. Clearly the sunshield can be very well thermally insulated from the rest of the telescopy. The sun-facing side is up to 300°C hot. The cold side is -233°C.
I'm not saying radiating the heat from the GPUs is a solved problem. Just the thermal insulation between the solar panels and the rest of the spacecraft.
