I hate to put even arbitrary numbers out there because folks tend to fixate on the numbers instead of the exercise itself.
The purpose of a private beta is to show that:
- The topic has enough interest to succeed
- The topic works pretty naturally within the framework of objective Q&A
- High-quality, original information about the topic is abundant after a short private beta period.
It's the third one that requires a bit of interpretation on our part. If a site has 150 questions with answers that just paraphrase Wikipedia, that site has a problem that would probably cause it to (at the least) be held back in private beta for a while longer.
If a site only manages to get 40 questions but they all have fantastic, in-depth answers, then we'd be very inclined to give it more time and see what happens.
If a site only manages to get 30 - 40 questions, and the quality is just not that great, then we'd probably be letting them know that it just didn't work.
So really, the best advice - shoot for as much high quality information as you can. Don't try to hit an arbitrary number of questions, because what you'll end up doing is just diminishing the overall quality of the site through deliberate content generation rather than asking actual questions about problems that you have, or recently had.
Any quality issues? Any signs that they ran out of things to ask? Any major problems getting the topic to fit our Q&A framework? - these are the most important questions we ask when we evaluate a private beta. The answer to these questions should be "no!"
Did they manage to hit (arbitrary) number of questions? Do you think they could if we gave them another week? If the answer to either of these is "yes", then you go public, or get a bit more time.