Interesting thought.
IANAL, but I understand that most successful prosecutions have been for /distributing/ copyright material, rather than /consuming/ it. I think that’s quite a big distinction, legally, and watching a dodgy recording is less clearly illegal in many countries and may be entirely legal - even if they could prove that filename = actual file.
It’s fine, but not going to be the cheapest.
Cheap to buy: Any old PC desktop, really. Most will run linux and windows fine, depending on what you want. Anywhere from free to £100. If you have an old desktop or laptop already, use that to start with.
Cheap to run: Any mini PC. I run a Lenovo ThinkCentre M53 for low power duties. Cost £40 and runs silently at 10watts, idle. (I have a secondary, much beefier server for other stuff that runs at around 100w which lives in the garage)
But plenty of people do run mac minis as home servers, often on Linux. They’re fine - just do your homework on the CPU ability, how much ram you can add, and whether you’re okay with external disks if you can’t fit enough inside.