You can grow a tremendous amount of food in a relatively small space. Square foot gardening is a good example of this (http://www.squarefootgardening.com/). The trick is in crop selection - if you grow things that do well near each other you end up with fantastic yields. The common example is the "3 sisters" method of growing - melons + corn + beans, for example. The melons provide a shade canopy at the soil level, the corn grows tall so it gets enough sun and the beans use the corn stalks as a trellis.

Of course, this isn't a method that scales up at all; machines can't be used in harvesting since you have mixed crops everywhere. Arguably they wouldn't be so necessary for community gardens though.