You need to host Jellyfin on your own server. For example, you can get a micro pc and attach a hard drive to it with N terabytes of your legally acquired movies and tv shows.

Getting it working right is a pain (I keep running into problems where it stops working, or the metadata is wrong, it forgets how to access the drive, and so on). You also need to ensure it has an IP address to access it from afar, and set us SSL encryption if you want secure access from afar. Expect to put aside several hours to set it up, and then several more to fix it when it breaks inexplicably days or months down the line.

Then you can connect to it from your browser directly or via the jellyfin app. Apple TV even has a jellyfin app, so I can watch from my parents’ TV setup. You can set things up so that different users can access different shows even.