Central computer with terminals in the rooms

By the way, local backups don't help you much if your house burns down... = Vote for the cloud (which doesn't have to be foreign)...

Thank you very much for the great answers!

This is not about "needing". I find it completely okay. You have to see it as a hobby.

That's how I see it too :D

PC for 2000 and 3 monitors doesn't say anything about the requirements.

So my requirement is that everything works. Also graphically demanding tasks. No gaming but that is equivalent so I say gaming should also be possible in this case. :)

It's about latency for me. If you don't need it super low, you can solve everything remotely, although multiple monitors are really difficult (I don't have a solution for that either). If you need very low latency (reaction games, e.g. shooters, in the worst case also online), remote is not worth much in my opinion. Then you have to solve it with video cabling. You probably need an HDMI repeater for that (at the latest from 10m).

The latency is not important. What is important is that the full performance (including graphics performance) arrives at the other end and the image is displayed on 4 monitors (also with different resolutions).

Hedgehog ud7 ;) No idea what client and software cost. Couldn't you use a soundproof rack? Or in the room next door? Even powerful computers can be made quiet. You don't need a gaming PC. With SSDs, large slow-spinning fans, it will be pretty noiseless. A €2000 PC is not really anything special. Powerful computers/workstations quickly cost five figures.

It is important that the full performance (including graphics performance) reaches the other end and the image is displayed on 4 monitors (also with different resolutions).

Then a direct connection to the computer remains probably the appropriate option. Plan a small room next to your office as a "server room." Then you can go directly from there into your office and – if really desired – enable access from other stations remotely. Data e.g. on a separate data server. Whereas I absolutely cannot see the advantage of such a constellation compared to "the computer is in the office." Noise cannot be decisive. You are merely incurring considerable costs without added value.