Mini pcs from aliexpress. Go for 12th gen intel ones

The 1 liter size offerings by HP, Dell, or Lenovo (all secondhand) are nice for this. 8th gen Intel options will be $120-$160 on eBay depending on if you need to upgrade them from base (usually 8 GB RAM and 256 or 512 GB nvme). If you get the 35 W editions (usually those with *T processors) you'll have something that also runs nice and cool.

The lines to look at:

• HP: Elitedesk 800 mini
• Dell: Optiplex micto
• Lenovo: Thinkcentre tiny

If you look at the same gen of processor they're virtually identical in most ways that matter. They do each have their own slight advantages over one another though. HP ones tend to have more storage expansion, are easier to open, and have better build quality (imo). Lenovo ones often have a funky little pcie slot that needs a riser to use, but it's the only one with this option. Dell ones are the most common and will have the most examples for teardown and maintenance. Dell ones will likely have the best firmware support.

They all work great with Linux, too. If you're downloading stuff, you'll want to make sure your download client can only access the internet via a VPN, which is easier to do with Linux via docker containers or a virtual machine.

Oh and for WiFi and Bluetooth: I believe all of the 8th gen+ options have an M.2 slot for a wifi+BT card and little antennas you can attach to it. You'll want to verify this for any specific model you're interested in. I bought and then returned several of these a while ago and ones from eBay would sometimes randomly come with a card. Intel ones cost about $15 on ebay. You can also use USB ones but they're usually less effective. Some ebay sellers will include a USB wifi+bt dongle with your order.

find a used NUC, or a mini-pc like a dell optiplex

Make sure whatever you get has native hardware decoding for whatever formats of video you download, otherwise youre going to struggle with watching video.

I snagged a Dell OptiPlex 7050 Micro from eBay for around 90 bucks, so something like this: https://www.ebay.com/itm/115785371324

I don't know about Bluetooth, but it meets your other requirements. The only con is the PC I got didn't come with an antenna, but I just used an antenna from an old network card and it seems to be working fine.

You can typically find the little refurbed dell Optiplex towers for that.

The "micro" models are very small and would likely work

Used 1L PCs are great for this

I use an asus cn62 chromebox for my jellyfin. A Celeron model. Got it for like $30 off eBay with a power cord and then upgraded the ram and SSD.

Not sure about the Bluetooth and audio out tho. But thin clients or an old sff dell optiplex is probably the way you wanna go.

Lots of people have already recommended secondhand compact PCs, like the kind most public libraries use to provide public computer access. As an alternative, if those are hard to come by in your area, I would recommend finding an old Lenovo Thinkpad. Even thinkpads as much as 10 years old remain very serviceable, significantly more powerful than most single-board project computers, and their power supplies only draw about 65 watts maximum, much less than the average desktop power supply. I have a few I've collected, and one sits under my router permanently as my personal server. In fact, I'm sending this message from one right now!

Raspberry Pi 3-B is $35. You can get a half TB micro-ssd for $30. Plenty of cash left over to splurge on a fancy case off etsy if you want.

You could get the latest, Raspberry Pi 5, for $80 and still be under budget but they removed the 3.5mm and HDMI port in version 4 for a single usb-c out that can do AV.