Dance clubs, and to a lesser extent bars, are geared toward extroverts. For an introvert, they're downright hostile—they're loud, packed with strange people, and the expectation that you're going to get to know one or two of them. But I wonder if there's a way to make a club that caters to introverts. And if you made it, could you get people to come?
What group activities do introverts have? Board games at hobby shops, and Magic: The Gathering at the same. In areas with internet cafés, online multiplayer games paid by the hour. It seems that the common thread amongst introvert-friendly group activities is that they're all things that you have to have another person for, at least to get the same experience. (Single-player video games are very different from multiplayer, even in the same game.) They're also things that work in small groups, even when playing within a larger group.
So I guess an introvert club would want to have gaming rooms. Instead of a bar, maybe a coffee house or café area. The place would have to have free wireless internet.
The biggest problem with the idea besides the fact that there's no guarantee it would be successful at all is that it would almost certainly be far less profitable than a regular dance club. An introvert club would require many times the space of a regular club to comfortably house the same number of people. Most introvert activities don't mix well with alcohol, so the place would have to come up with a model where they can sell non-alcoholic drinks and overpriced snacks to make up for that.
I have my doubts that it could be successful, but it would be interesting. Even Lincoln, Nebraska could get together a few hundred people at a HobbyTown USA to have Magic tournaments on Saturdays; I'm sure a large metro area could offer enough people to have an introvert-focused club. You'd just need to convince people to go. Maybe it would just have to open a couple times a week at first; introverts seem to prefer their limited social activities to be planned out in advance. I have my doubts that it could work. But at least in concept, it would be nice if such a place existed.