I initially voted to close this question as too broad, but then I decided to answer in the negative. There is no general formula.
At 8 days, you aren't talking about killing an existing community. Rather, you're talking about not giving time for a community to form.
Factors to consider include:
- What kind of community is it? How much of a critical mass does it need to be sustainable? For example a wiki can start with a single person maintaining it if the topic is sufficiently narrow, but a chat room is only worthwhile if it has multiple people in it at the same time sufficiently often.
- What is the community about — how many potential users does it have? Have you tapped into the potential?
- How do you attract new users? What can the burgeoning community offer to them?
- Are there alternate communities? If so, are they complementary (e.g. a wiki and a discussion forum) and do you hope for people to join from the other one, or are they competing and are you relying on disgruntled users or a different type of audience?
In the case of Stack Exchange and the private beta to public beta milestone, the administrators have built experience from about 130 sites that passed that milestone, and a smaller number that failed. Through this experience, they have an idea of what's worked before and what hasn't. In particular, they have a good idea of how the next step — a public question and answers site — works.
- In order to attract people who are asking questions, there needs to be a minimum of content. A private beta typically comes out with at least ~100 questions, preferably more; less than that, and the site looks like an empty shell and doesn't attract people looking for a response.
- The private beta is a time to establish some moderation guidelines, which experience shows are needed on a Q&A site. Typical considerations include deciding what's on topic, working out how to classify questions, and establish quality norms.
- Sometimes, when there are already Stack Exchange on a related topic, the new site needs to establish itself as bringing something new to the existing offer.
When reviewing a private beta, there are three possible outcomes: move it to the next phase (i.e. go public), shut it down, or let it run longer. Going public is only possible if the criteria above are fulfilled. Deciding to spend a longer time in private beta is based on the expectation that the criteria will be fulfilled: a private Q&A site is not a goal in itself.
If experience shows at that point that the community isn't taking shape, then there's no point in letting it run longer. At some point, you have to decide to let it go. The user pool for a Stack Exchange private beta is about 200 people from all over the Internet who expressed interest in the preceding months, out of which maybe half will come. A week gives enough time for most of the people who would turn up, to turn up and participate, for the seed content to accumulate, and for the first discussions about the norms. If there's good content but too little, it might be worth trying longer. If the norms are still being actively debated but the debate is progressing, it might be worth trying longer. If the discussions are going nowhere or the content is desperately lacking, throwing more time at the problem won't help; the problem needs to change, or to be tackled by other people: in this case the best outcome is to close down the private beta and maybe try again later.
Generalizing this, you can have various milestones in the lifetime of a community. At each milestone, you have a rough idea, through experience, of what to expect in terms of volume, involvement, publicity, etc. If the expectations are met, you keep going. If the data is slightly below expectations, you give it more time and try to push the community in a sustainable direction. If the expectations are far from met, you shut down, or significantly reshape the community. There's no particular time window for these milestones in general: it's based at what you expect at that point in time.