Do you just use podcasts on your phone? If you have an Android phone, AntennaPod, while not self-hosted, works very well and is FOSS. There are other options to "self-host podcasts" to varying degrees:
PodHoarder: mentioned in another comment, which could be piped into AntennaPod, but I find that a bit redundant for me
AudioBookshelf: a fantastic self-hosted audiobook server, and an okay podcast server, but is focused around streaming from your server to your listening device, and I prefer to download on wifi to listen later (it was pretty clunky for that workflow).
GPodderSync: barely supported at this point and missing too many features to be useful in my opinion, but a neat backend for AntennaPod and other players to sync to some degree.
Bonus: the creators of AntennaPod and other FOSS podcasting software are working on a replacement for GPodderSync here: https://github.com/OpenPodcastAPI
EDIT: for RSS in general, I use FreshRSS, which uses the g-reader API to sync across multiple apps. It's awesome.