Yes, you will need a tuner card or USB tuner. The go to apps are mythtv and Kodi. I’d start with Kodi and see if that does the job for you. If you check the Kodi wiki you’ll find resources on what specific tuner cards work best.
Self hosted DVRs are definitely a dying breed, but there’s stuff out there if you’re willing to dig.
For Jellyfin integration, once you have the files you should be able to dump them out to a folder where Jellyfin can scan them.