It won't come as news that closet space is at a premium in New York City, especially if your family includes five kids. So when Avi and Michal Schick's two teenage daughters outgrew their old clothes, the Schicks did what a lot of people do: They placed them in bins in their Flatbush, Brooklyn, basement and invited less-fortunate locals to help themselves.

From that modest start grew Bobbie's Place, a Brooklyn children's clothing store that is conventional in every way but one—all the merchandise is free. Bobbie's Place serves...