Can I set a sleep timer with Podcasts?
Actually, the Podcasts app does have a sleep timer function. 

The easiest way to access this is to:


Open an actual episode (i.e. by double-tapping on it) so that it pops up the main episode screen (the pop-up with the episode artwork at top centre and the play/pause button below it with the skip buttons and volume slider)
Now swipe up from the bottom near the play/pause button (so that you're scrolling the page up, not bringing up the Control Centre)
Below the 1x button you’ll now see a Sleep Timer button (it has a moon icon on it)
Tap on the Sleep Timer button to access a number of options, including one to stop playback when the current episode ends


Below are some screenshots for reference:



Finally, there’s also an iOS-wide way to use a sleep timer. Not only does this work with the Podcasts app, but also any apps that play audio (incl. 3rd party apps). If you’re interested, see my answer here for details:
https://apple.stackexchange.com/a/289108/300323 . The gist of the solution is to use the "stop playing" option under "when timer ends".