ALL server farms radiate into space - they just use somebody else's atmosphere (or perhaps a river and the atmosphere) to convect the heat to the upper stratosphere, which is less infrared-opaque. Absorbing the energy Out There, processing the energy into computation, and radiating it Out There, is typically done very expensively and inefficiently, Especially if it is done with big hermetic boxes and circulating coolant.
We do that on Earth because chips are sensitive to contaminated air and dirty fingers. We crowd everything into small packages because of propagation delay, and because we learned to design software on uniprocessors, and use those same habits on parallel tasks.
A more-ideal space processor is a fabric with widely spaced compute nodes. The simplest way is a big surface with solar cells on the sunward side, with a grid of small "naked"chips on the back, radiating heat in towards 2.7 Kelvin deep space. A better way is a shape like a "T", where the top bar of the T is the solar array facing the Sun, and the larger area downstroke of the T is the compute fabric. The naked chips will be bathed in radiation - but the newest silicon processes bath them in 100eV extreme UV photons when the chips are made. There will be bit flips and an occasional fried transistor - but 10 nanometer transistors are a very small target for space radiation, So, design with error detection, correction, and redundancy. Design code for loosely coupled autonomous processes. Spread the grid of small chips over a area much larger than the solar cell, so that it can radiate heat at very temperatures. Rule of thumb, chip "wearout" lifetime doubles for a 10C drop in temperature, so why not aim for 200 Kelvin operating temperature?
If the sun-facing solar array reflects infrared and UV without absorption, and is 50% efficient with the 40%-of-1366W/m² midband light that it does collect, the waste heat is 275 W/m², and a 90% infrared-emissive solar cell back side would be 270K. The angular size of the Sun near Earth's orbit is 32 minutes of arc, about 0.01 radians, so the triangular dark shadow umbra behind the solar cell is 50 times longer than the solar cell is wide. In theory, a two-sided triangular chip-covered "pennant" in back could have a Stephan-Boltzmann law radiating area (both sides) 50 times larger than the solar cell, hence 40% of the temperature in Kelvins of that solar cell - perhaps 110K - but it would be cheaper to make the heatsink blade thinner, hotter near the chips and colder between, with a truncated blade. In conclusion, a properly designed solar powered computer segment will need no plumbing or coolant if the computer chips are small, numerous, and distributed.
Another emerging technology is fiber-optic chip interconnect. Optical fiber fed by wavelength division multiplexing/demultiplexing etalon chiplets can move Tbps data over a 125nm fiber optic core, with practically zero path loss over subkilometer distances. My 300/300 Mbps fiber internet link is all optical from my house to the switching center 20 km away, nothing to depower or burn out. A 10-meter-scale compute surface could have tens of thousands of short optical fibers imbedded in a heat-sink "sandwich" - a few grams per square meter.
Chip size? More than a decade ago, I licensed circuit technology to Hitachi ULSI for their "smart dust" RFID chips, 300x300x60 micrometers. Using the "fluidic self assembly" techniques developed by Alien Technologies (cool name!) of California, we could have made RFID product tags cost less than a paper price sticker, and saved storefront retail. Then came the Sendai earthquake; Hitachi shifted corporate focus to national recovery, and Amazon engulfed retail.
Orbital location? 1400 km altitude (7778 km equatorial radius) Low Earth Orbit (MEO) is SILLY; the orbit is in shadow 30% of the time, 2000 seconds every 1.9 hours. Besides the downtime, thermal stress could be a huge problem.
How do I know? I spent many years designing gossamer "thinsats" for a past brainfart, http://server-sky.com . Thermal shock becomes less problematic if the gossamer "thinsats" turn sideways approaching the penumbra; this also reduces light pollution in the Earth's nighttime sky; Unlike SpaceX Starlink, I like astronomers.
My current thinking is to move the arrays in front of Earth-Sun L1, Four space probes there now (SOHO, Wind, ACE, and Aditya-L1) in "Lissajous orbits" near the L1 gravity cusp in space, 1.5 million km sunwards; 10 second round trip signalling time. Continuous sunlight, no thermal shock, very small station-keeping thrust - which can be created by "solar sailing". There is room in that region to collect almost 100 Earth's of sunlight without reducing Earth's illumination by more that 1%.
And frankly, I prefer that humans to have a wee bit of terrestrial reaction-time advantage compared to exawatt AI.