lightsystem

The plugin is configurable letting you define what feeds to read and how often to read them. It also lets you decide whether you want to store the external images locally, inserting them into the media gallery, or otherwise simply hotlink them (that is the default).
You will find these settings in your administration page on the Settings->RSS Settings sub-menu.
Check out the <a href="https://github.com/LightSystem/WordPress-Plugin-RSS-Sync" rel="nofollow ugc">project on GitHub</a>.

RSS Synchronisation

Use this plugin if you wish to read external RSS feeds into WordPress posts.

SecondLineThemes

Sync Podcast RSS feeds with your WordPress website automatically. The Podcast Importer plugin helps to easily import podcasts into WordPress. You can import your podcast into the regular WordPress posts or into a custom post type (if you have an existing one). 
The plugin fully supports popular WordPress podcasting plugins such as PowerPress, Seriously Simple Podcasting, Simple Podcast Press, and works even better with podcast themes developed by <a href="https://secondlinethemes.com" rel="nofollow ugc">SecondLineThemes</a>
The plugin supports importing episodes into existing custom post types, assign categories, import featured images and more. Additionally, the plugin enables continuous import or “Sync” of podcast RSS feeds, so every time you release a new podcast episode, it could be automatically created within WordPress. You can also set multiple import schedules and import different podcasts from separate sources at the same time. (For example, when importing separate podcasts from separate feeds into one website)
To use the plugin, simply run a new import under “Tools -> Podcast Importer SecondLine” via the main menu that appears in your WordPress dashboard. Set the different options and if you need a continuous import process for future episodes, make sure to hit that checkbox before running the import process. 
You can disable a schedueld import at any time by simply deleting the import entry.
The plugin also supports automatic import of native / embed audio players from 15+ podcast hosting providers, including: Buzzsprout, Megaphone, Pinecast, Captivate, Transistor, Spotify for Creators, Simplecast, Podbean, Whooshkaa, Omny, Ausha, Spreaker, <a href="https://beamly.com" rel="nofollow ugc">Beamly</a>, Audioboom, Fireside, Libsyn and more.
<h3>Pro Version</h3>
The Pro version can be found here – <a href="https://secondlinethemes.com/podcast-importer-pro" rel="nofollow ugc">https://secondlinethemes.com/podcast-importer-pro</a> 
It includes: 
* Unlimited scheduled imports for podcasts/shows. 
* Import to any Custom Post Type or Custom Taxonomy. 
* Set specific import intervals / times. 
* Import transcripts from RSS feed. 
* Import audio player to custom fields. 
* Import tags and categories from the feeds. 
* Force a re-sync on all existing episodes (to update metadata) 
* Set a global featured image to all imported episodes. 
* Manual “Sync” button to sync on demand.
<h3>About SecondLineThemes</h3>
SecondLineThemes is developing unique WordPress themes and plugins for Podcasters. Our tools are very popular among podcasters. To hear more about us please check our website: 
<a href="https://secondlinethemes.com" rel="nofollow ugc">https://secondlinethemes.com</a>

Podcast Importer SecondLine

An excellent plugin for importing podcasts into your WordPress website.

The only downside is that there is no option to limit the number or age of the post on the initial download. Can be a bit out of date when you're downloading podcasts from 2011 and earlier.

November 2021
Downgraded rating from 5 stars to 3 stars due to latest update placing in nudge spam on every admin page to use their theme.

Excellent plugin for importing podcasts

A simple podcast import tool for WordPress.

RSS Synchronisation vs Podcast Importer SecondLine