There's an answer here on Meta Stack Exchange by former staff member @TimPost. The first sentence is the most important:
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.