If your wanting to get into it as cheap as possible, here's how I'd do it, in stages

Use an old desktop. Install some Linux based is. I really like truenas but there's heaps of options. Proxmox has been mentioned too

Install Plex, but I encourage you to install jellyfin as well and point it at the same library. Just try it out and see how you like it.

Think about how important your media library is and if you want some redundancy but honestly most media is easily replaceable