Probably something 7th gen Intel or newer so you can use Quicksync for transcoding on Jellyfin and HW accel on Immich for ML, face recognition, resizing, etc..
Tons of 7th/8th gen PCs pulled from offices on ebay for around $50-80, if you do some creative mounting inside a Midtower (MT) sized one you can fit a couple 3.5" drives and 2.5" SSDs.
On the budget side there is:
raspberry pi + usb RAID enclosure that can spin down the harddrive (I used icybox in the past)
Instead of a pi, it is also interesting to look at mini PCs like N97 or N100 which are super cheap nowadays, about the same as a rpi5 if you consider whats included / what addons you'd have to buy for the pi. Mini PC have more RAM, a build in nvme, a way beefier CPU. But their power efficiency is way worse.
Depends on how much horsepower you need or how energy efficient you want to be.