I run cables through the walls. I have a computer room with various DAWs, backup systems, servers, etc., and all you have to do is drill an appropriate hole in the wall (do this safely, be careful you don’t hit anything important! Do NOT screw that up!), then I run the needed HDMI, DisplayPort, USB, etc. cables into the main studio. Just make sure to buy very high quality cables to get reasonable lengths. YMMV.
I over-spec all the cables and keep them as short as possible. I’ve bought tons of cables, quality varies significantly, even from the same company. You can use tricks to get longer runs if needed, for example with powered hubs, run lower resolutions or refresh rates on over-spec’d cables, etc… (4K 120Hz will require shorter cable runs in my experience than, say, 1440p 60Hz, and so forth.)
Has worked great for many years. The only time I bother to put computers in the control room is when I’m building the machine, troubleshooting, or doing something non-critical. I use sound dampening cases, Noctua fans in everything, including the CPU coolers, so they are still pretty quiet, but to get that truly quiet, you have to move the machine to another room.
The one exception to this is Apple Silicon (M1, M2, M3, M4, etc.) machines, which are ultra quiet. I have recently decided to phase all those out, though, but my Apple Silicon machines can easily be installed in the studio with no issues for noise. I have recently decided to dump all Apple products though (for various reasons), so my Windows and Linux machines are in other rooms.
Again, YMMV, you need to be very careful if you’re drilling holes. If you cannot drill holes for some reason, then you might still be able to run cables around the corner under a doorway if there is clearance. If you are nervous about drilling, maybe hire a contractor to do it safely. PLEASE do it safely. Then just do the math on the cable runs, buy the best cables you can, and your noise levels will be completely solved.
IMPORTANT: you may need to reduce resolution/refresh rate to achieve cable length. And please be aware some devices, computer monitors, etc., are more sensitive to signal degradation (i.e. cable length) than others. You will need to do some serious testing to make sure everything works correctly to find the maximum length that device will work at, then err on the shorter side for safety. High bandwidth devices like hard drives CANNOT reliably be placed on long cable runs, for example. Keep those close to or attached to your computer. Again, test everything, one by one. I’ve found some devices work fine up to around 20 feet (6 meters), but others are very sensitive at 6-10 feet (2-3 meters). I have one Acer monitor that works at 25 feet no problem. Another, different brand (Samsung), did not like 25 feet, and I had to settle with 15 feet for it.
And again, there are workarounds, like HDMI repeaters, powered hubs, etc., so you can actually go farther than that if needed. Just test, test, test, test. Be prepared to buy lots of cables.
And with some strategic layout of your studio, you can configure your spaces for shorter cable runs… It took me years to get my space(s) set up properly, but it’s totally possible. Lots of trial and error though.
Also be aware that your machines still need proper airflow, so don’t put them in a small enclosed closet! So the “room” where you put them needs to be appropriate!
Cheers and good luck! And again, YMMV.