Linux

You are making a mistake.

Do not build one big computer to do everything.

The system requirements are wildly different for the different things you’re asking. That means your computer can’t make compromises to be cheap.

Jellyfin and the piracy stack will want a potato with middling amounts of ram and a cheap discrete gpu from six years ago. And whatever storage you plan on using.

Minecraft will want ram and fast cpu.

Streaming will want a recent cpu with all the bells and whistles and a remote.

Playing games needs a okay gpu.

When you combine all that you need an okay gpu, a recent fast processor, a good amount of ram and the rest of the stuff to make it a computer. Plus your time. That’s approaching $800 now.

If you split it up you can do jellyfin, the arrs and Minecraft server on a ten year old gaming pc of yore that someone thinks they screwed you on when you pay them $100 for at a yard sale and hook your tv to a dedicated streaming set top box that can talk to your jellyfin for $100.

Plasma Bigscreen is kind of a mess still, so maybe run it on something first to understand what you're getting into. That being said, any distro that runs KDE can run KBS, but will probably not be tuned for the least amount of resources to be used.

Something like OpenELEC/LibreELEC or OSMC are very well tuned for this, and require very little in the way of resources though. I'm not saying DON'T use KBS, but if you're lacking in resources, the others are probably a better choice.

any recent-ish intel machine should do, i'd suggest around kaby lake or newer for the igpu's hw decoding capabilities. beware that netflix streaming may not work so well with linux and stuff because of shitty drm.

I guess you might want to ask this question in one of the self-hosting communities (there's a bunch that are active, so I'm not linking one)

But getting back to the topic:

• jellyfin needs very little
  I'm running mine on a laptop with 4 threads at 2.6GHz and on 8GB of RAM it was fine with a bunch of other stuff running
• OwnCast
  IDK but for personal use I'd expect to also have low requirements
• Minecraft server
  That part I'm not sure about. I'd expect it to be a bit more demanding
• play some steam games
  I understand you'll be running those on another computer and only streaming via OwnCast, so that goes back to earlier point

You’ll want a GPU with hardware encoding and decoding for popular codecs. Most GPUs these days do that, even the budget ones, but you do need one.

Minecraft servers want fast single core CPU performance and a decent amount of RAM.

The rest of it could be RAM hungry too.

Minecraft server on Java implementation super hungry for ram

Bigscreen only just came back around, I wouldn't go too heavy there yet (and I'm testing it out already myself)

Get flex launcher instead. Simple config, does the job.

In terms of specs, there is a lot in your list that can vary, and make those specs jump. Where to start, IMO, would be a 7th gen Intel. If an 11th gen is an option, there and up transcoded av1.

For storage you're going to have to do the math yourself in terms of what you'll have. I have 40tb of media. I used to have a 1tb drive for everything. Storage is how much you want to allocate, nothing more or less.

Ram is similar. Find the worst case game, most overloaded with plugins Minecraft server or whatever, and start there on memory use. JF won't take much, your game choice might.