<itunes:block>Yes</itunes:block> tag to a feed will instruct complying podcast directories (such as Overcast or iTunes) to not list that feed publicly. I also found mentions of a
<googleplay:block> tag, but even Google's own docs only seem to mention the iTunes tag.
I haven't found an equivalent for JSON Feed, but I'm still not sure how widespread support actually is, especially among podcast players.
I'm only writing this up because it was strangely difficult to find any useful info when I initially searched for things like "mark podcast feed private" or "rss private tag". Most of the results were from paid podcasts hosts with no technical details.