This seems like a perfect use case for a small home server. If they find a revenue stream then they can scale to renting some rack space somewhere. I know its not as easy as clicking a button in AWS but its not totally debilitating.
As far as hosting, it can certainly be done privately, but it requires more work than cloud hosting, as gab learned when they went entirely internally hosted.
Considering how StackeOverflow ran on a small handful of machines to a pretty large scale (and still isn't excessive), I can't see how more than a few machines should be necessary to run a site that serves small texts, even among millions of users.
There is one caveat though and that's media. If media is transcoded, and/or if more than a small fraction of messages contain video - then the service might face using most of its resources towards that. But in that case , if I was running their ops I'd just disable media attachments while figuring out how to run the full service.