It was founded by archivist types. They don't take things down without cause. Actually, they frequently don't take things down with cause.
archive.org and the wayback machine are also good options.
Unfortunately, if you're not willing to pay for hosting, you can't guarantee stability in your website: free hosts are rarely the most economically stable. Neocities is currently the best option there, but for how long I don't know. The good thing about a website though, in contrast to google, is that if you make sure to use a relative path structure when you write it you can just drag and drop it onto any webhost and it will work. Google docs is more locked to Google's ecosystem. Exporting is possible, as is, through the publish to web system, getting basically a website (which you can then download and modify until it can be distributed via any webhost...). I think the google publish to web links will actually survive this (browse to one in incognito mode after all this goes down and see...), backing up those is insurance. But passing around raw docs links is fragile.
Generally speaking, Google has a habit of discontinuing services, though. I don't trust them. Although I'm pretty sure Docs makes too much money for them to get rid of it right now, I wouldn't be shocked if publish to web eventually goes away (although, who knows, maybe it won't).
In short, no, there is literally no such thing as a stable host. Servers go down, websites crash, companies go out of business, multimillion dollar conglomerates decide they can't be bothered to support your usecase.