My media server, which is just my server generally, is an old thinkpad I have from 2014. For media I use Jellyfin and I ensure the content is already in a format that will not require transcoding on any device I care to serve to (typically mp4 1080p hevc + aac).

If you look at the used computer market, there are endless options to attain what you are asking for. My only real advice is make sure the computer doesn’t draw much power and, if possible, doesn’t emit much or any fan noise. A laptop is a decent choice because the battery kind of serves as an uninterruptible power supply. I just cap my charge limit at 80% since I never unplug it.