You can use that site to help you find compatible parts within your budget.
I'll throw out an example build but I did 0 research into these parts so don't take it as gospel just a very general idea of what it should look like. I'm sure you can get it even smaller and definitely cheaper. I have quite a bit of experience for a laymen doing this. But I've never built something that approaches the form factor of a mac mini.
Here is the link to the sample build
https://pcpartpicker.com/list/qVkkbh
That was something i literally just clicked through, there are probably better examples of small form factor stuff. I'll go through each component and give the choice some context then you're on your own.
CPU - i3, anything modern will do here but the i3 has very good single core performance and should shred most of your media center needs. If you're doing very basic media server functions going past this should see basically no real gains for you beyond maybe compressing files. Could definitely go much cheaper here and lose very little. But at 100 dollars for the centerpiece of the system why not
Cooler - just a noctua with a more flat form factor so it fits in everything easier. Depending on the case you could pick a larger one. Noctua makes very quiet fans but they are ugly. Many other options on this one. Again can go cheaper, important part for the heat sink is the fan. That's what will make noise and why i would say buy a nicer cooler, almost purely for the fan
RAM - 8g is probably all you need if it's just a media server. Could double and go to 16 gb, could also get ram with better response. But neither of those things should improve performance in any substantive way
Storage - just put in a cheap SSD, could get other form factors or even use a USB stick as your main drive. Many options here. Works very similar to mac
PSU - this one you could go much cheaper. The be quiet i'm sure will be as the name implies relatively silent if the fan is going. But you might be able to go much cheaper and get very similar results.
edit - case - this is the one most determined by taste. You can spend a lot or a little. Whatever you do, do not cheap out on the fans if it being silent is important. i just picked a random cheaper one that wasn't rosewill
Now these are just my thoughts. Google "makings a windows pc like mac mini" or something. And look through that PC part picker site. I'm sure tehre are "small form factor" builds out there you can just 100% copy so you know everything fits.
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