One of the closest approximations was in parts of Spain in late 1936-early 1937. The CNT, an anarchosyndicalist labor union, was active in the fight against Franco, and local organizations had widespread support.
A longer list appears on Wikipedia:
http://en.wikipedia.org/wiki/List_of_anarchist_communities
In the Spanish case, this collapsed due to (1) military pressure from the Nationalists and (2) treachery by the Republic. The Republic needed Russian arms supplies, and Stalin threatened to cut these off if the Republican government didn't get rid of the CNT.
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Anarchism supports free association, opposes the state, and opposes hierarchy. (At the very least, anarchists oppose involuntary hierarchy; anarchists often try to create egalitarian alternatives to the semi-voluntary hierarchies too).
There are several different traditions which respect these values, and usually recognize each other as forms of anarchism. These traditions borrow ideas from classical liberalism, from early socialism, from each other, and sometimes from other sources.
People depend on each other. People tend to create their own voluntary social order, including free association, reciprocity, mutual aid, and, if necessary, mutual defense. Once people create this order, a state, or any other criminal gang, is in trouble. So the state, to preserve itself, must preempt voluntary social order.
Highleyman, "An introduction to anarchism:"
http://www.spunk.org/texts/intro/sp001550.html
"An anarchist FAQ:"
http://www.geocities.com/CapitolHill/1931/ or
http://www.infoshop.org/faq/index.html (same text)
A messier source:
http://en.wikipedia.org/wiki/Anarchism
Many other anarchist works are also available at:
http://dwardmac.pitzer.edu/Anarchist_Archives/ and
http://www.infoshop.org/library/Main_Page (different works) and elsewhere