ALL server farms radiate into space - they just use everyone 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 a service like communication or observation, and then radiating it Out There, is typically done very expensively and inefficiently. Especially if computing is done with big hermetic boxes and circulating coolant.
On Earth, we put computation in boxes because chips are sensitive to contaminated air and dirty fingers. We crowd everything into small packages because of propagation delay, and also because we learned to design software on uni-processors, using those same habits on parallel tasks.
A more-ideal space processor is a fabric of many tiny compute nodes widely spaced to radiate heat. 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 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 panel facing the Sun, and the larger area down-stroke of the T is the compute fabric. The naked chips will be bathed in radiation - but the newest silicon manufacturing processes bath chips in 100eV extreme UV photons when they 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 software 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 low temperatures. Rule of thumb, chip "wear-out" lifetime doubles for a 10C drop in temperature, so why not aim for 200 Kelvin operating temperature, instead of the 370 Kelvin CPU temperature in your tower PC?
If the sun-facing solar array can reflect useless infrared and (perhaps) UV without absorption, and is 50% efficient with the 40%-of-1366W/m² mid-band 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. However, it would be cheaper to make the heat-sink 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 9μm fiber optic core, with practically zero path loss over sub-kilometer distances. My 200/200 Mbps Ziply fiber internet link is all optical from my house to the switching center 20 km away, nothing to de-power or burn out. A 10-meter-scale compute surface could have tens of thousands of short μm-diameter optical fibers embedded 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-from-center) 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 night time sky; Unlike SpaceX Starlink, I like astronomers.
My current thinking is to move the arrays in front of Earth-Sun L1. Four space probes are there now (SOHO, Wind, ACE, and Aditya-L1) in "Lissajous orbits" near the L1 gravity cusp in space, 1.5 million km sun-wards; 10 second round trip signalling time. Continuous sunlight, no thermal shock, no "warm Earth filling very small station-keeping thrust - which can be created by "solar sailing". There is room in that region to collect 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.