There's a few different things people do but the most common thing is to run a stack of the "servarrs". These are services that automatically find, track, and download movies and shows. Mostly this is on request. You get various search interfaces to find stuff that's out there in the TV world or the movie world or the music world, you request it, and your server uses bittorrent or usenet or both to download the parts that it needs. Then it's held on a hard drive until you watch it, at your leisure, and keep it or delete it.

I think peoples' preferences for how long they keep things and how they find new things to watch are very personal, but the stack gives you almost too much flexibility about it.

Since you mentioned cost, this will usually run to a few hundred dollars for hardware (including the drives), but I think you could scrape together something barebones for less than a hundred and barely notice the difference.