@timapple Most podcast players support downloading episodes for offline listening. So as long as the RSS feed contains all episodes, that's probably the easiest way.
I tried with iCatcher! on my iPhone just now. I didn't download hundreds of episodes, but the app makes that easy (you can choose how many to fetch). The files end up in On My iPhone > iCatcher! > Podcast Name. From there, you could move the directory to iCloud or share it via AirDrop or whatever.
The workflow will differ depending on which app you use.