In short, I think Eric had it exactly right when he told me early on in the development of Pumping Station: One that a hackerspace is a YMCA for geeks and artists.
A hackerspace is an interdisciplinary community for learning, teaching, and creating. Instead of starting with a defined range of projects or programming, a hackerspace is driven by its members. It is a place where members have the infrastructure and resources to work on projects that interest them. Hackerspaces promote people to be hackers in the broadest sense: to learn all they can about the fields that interest them, explore their bounds, and create new and interesting ways to apply that knowledge.
The people in a hackerspace also share their knowledge with others who share their interests, through classes, working groups, or day-to-day discussion while working on projects. That is where the fascinating educational potential of the hackerspace lies: there is no finite list of the skills that can be taught and exchanged. People share what they know with members and the community at large, and it results in more people having the knowledge to make something new and tangible out of their ideas and interests.
For some more perspective on Pumping Station: One, as well as the past, present, and future of hackerspaces, watch this fifteen-minute interview of Eric Michaud, president of Pumping Station: One: