robots.txt unreachable: Google search removed my site!
Every few months I remember that I have a blog, and then pop open the Google Search Console to make sure all my pages are indexed and see if people are finding the stuff I post.
When I did this in December I was quite startled to see that I was getting literally zero traffic from Google Search.

Fortunately, Google's Search Console told me exactly what was going on: in its latest pass on my site, it put the whole thing into an error category of not being indexable because it could not find a /robots.txt file ("robots.txt unreachable").
Now, this site has not had a robots.txt for years (which certainly explains why Google couldn't find one). All the pages are public anyway, and I assumed that crawlers would crawl by default unless they hit a robots.txt Disallow rule or a noindex header. So why bother adding that file?
Well, it turns out that not having that file got me into trouble: as far as I can tell, it's supposed to be the case that Google will treat an absent robots.txt as permission to index, but what does "absent" mean?
Apparently, a recent change to the SvelteKit Cloudflare adapter (used to host this site) made it so that URLs to non-existent paths return 500 Internal Server Error status codes in production instead of 404 Not Found, so my bet is that this is what caused Google to treat my site has having an "unreachable" robots.txt rather than an absent one.
In other words: If Google gets a 500 error on your robots.txt, it'll delist your entire site from Google Search!
On the one hand, it makes sense that Google would interpret a 500 error to mean that there might be a robots.txt, and therefore might be a rule that says Google can't index the site, and therefore must proceed with caution to ensure it doesn't index stuff it shouldn't.
On the other hand, if a site is already indexed, and Google can reach its sitemaps, and none of the pages include a noindex header... it really seems like de-indexing the entire site is about as wrong of a move as Google could make. Truly wild behavior for this scenario. Unhinged, even. And also counter to what Google says about the robots.txt:
A robots.txt file tells search engine crawlers which URLs the crawler can access on your site. This is used mainly to avoid overloading your site with requests; it is not a mechanism for keeping a web page out of Google. To keep a web page out of Google, block indexing with noindex or password-protect the page.
How to fix it?
If this has happened to you, take heart that it looks like recovery is possible:

Fixing it will depend on what exactly is happening. I'm still assuming that you don't actually need a robots.txt, as long as that URL returns a 404 status or something else GoogleBot likes. But why risk it? Add a robots.txt file if you don't have one! And otherwise, uh, just make sure that your server is super reliable about delivering it?
For my case, I added a robots.txt file and then resubmitted my sitemaps via Google Search Console. It took about two weeks to start getting some visibility back, and right now most pages are "Crawled but not indexed", which is a cryptic state that Google provides very little information about. I expect it'll be another few weeks before Google has my site fully indexed again (it's sure taking its sweet time), and who knows how long it'll take after that for my search visibility to get back to where it was.
At some point I'll need to figure out why I'm getting these 500 pages instead of 404s, but that's a problem for future me.
Do other search engines do this?
I've seen no disruption to search performance in Bing, but that's the only other engine I've checked.