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.
I currently use AntennaPod with NextCloud's version of GPodderSync; which is...adequate.
My ultimate solution would be one where a self-hosted app that tracks and download my podcasts, and then proxy them with some sync mechanism to an android app, but keep them as separate shows with artwork and stuff rather than consolidate them into one feed. I could then choose to listen on the self-hosted web interface or the Android app, and they would be kept in sync.
I wanted to follow up here because I became aware of a pretty good solution for this. It's called pine pods and it's an open source self-hosted service that also syncs with GPodder (or Nextcloud's version of it) so it will keep AntennaPod and itself synched.
Lol no seriously, what's your goal here? Self-hosting a server seems entirely unnecessary.
If you want to host an RSS server, FreshRSS is easy to set up if you know how to do Docker stuff. Then, you could connect it to a podcast app on your phone. But all that seems very complicated when you could just install AntennaPod (which is open source), subscribe to a podcast's official RSS feed, and turn on notifications for that podcast. Adding an RSS server between your listening device and the original RSS server is unnecessary IMO, unless you have a use case that I'm not understanding.
I am really happy with miniflux. It supports mobile and has good keyboard navigation support on Desktop. It's also runs really smooth on my rather low powered homeserver (rpi 2b).
For push notifications you can use a simple script with ntfy. For example miniflux-ntfy.
Not sure if it meets your needs but I’ve been using Audiobookshelf for pulling podcasts. The default web interface is simple and straightforward and you can create secondary rss feeds if you have another podcast app you prefer. Has apps for iOS and android, though the iOS app is TestFlight. As a podcast player it’s decent. Not sure if it does notifications, I tend to disable them.
I started using it because one of the podcasts I listened to ended and I wasn’t able to go back and relisten to the episodes. Decided to start archiving the podcasts I was interested in.
Podhoarder is nice but more geared toward hoarding and a bit complex for listening.
Podfetch is nice and currently maintained but has some issues with proxy auth. It might cut the cheese for you I think. It has a weird naming scheme on disk tough.
AudioBookReader is amazing and still currently under very active development. With its mobile app is perfect for my use case. Podcast support is just fine for me.
PodGrabber seems abandoned since 2022, but I didn't try it.
For podcasts I recommend audiobookshelf. It has a very nice podcasts category with ability to backup the podcastst yourself, also syncs progress over multiple devices.