Maybe Audiobookshelf? You can set up podcasts and stream them from the web interface. It’ll keep track of your progress, so you can switch from web to phone and pick up in the same spot.
Edit: though it does download the podcast to the server so that you can stream it, but it grabs them automatically.
Some of the recommendations I have seen are Baikal and Radicale. I don’t think either have an official Docker image, but there are a few community-maintained images.