https://github.com/advplyr/audiobookshelf

Stores all your podcasts on the local computer you're running it on, uses an iOS or Android app to allow for remote streaming of those podcasts, and even downloads them to your device for offline listening. I think you can even turn some collections into podcast feeds or share the entries as their own podcast feed, or more simply, you can give people accounts on the system and they get access to the downloaded podcasts too.