I do not believe human behavior would allow such a system to exist for long without a clear advantage for the citizens of the anarchist area. Somewhere, someone is gong to think, "Hey, that's such a nice place over there. The people rule themselves....it's about time they learned about a REAL leader!" and up comes an army built through threats, bribes, and favors. These brutes would constantly threaten the community, and could one day overpower the anarchist movement.
That's not why I think it's unlikely, though. Inside the anarchist community, some people will either feel greed or over-entitlement. They'll think, "Why, these people need X, I'll get them X, and they'll respect me. I'll make them rely on me for X, so I cannot be replaced or overthrown!" or, "I am completely free here, right? So I am free to organize my friends and gather support for "defense", right?" Humans don't do well in an environment where the "fair share" relies on everyone's ability to speak up for themselves and pull their own weight. When any sort of competition is involved, we fight. We fight hard and sometimes we hurt others to make sure we get ahead. Until Humanity conquers it's competitive, selfish nature globally, we can never see a successful anarchist society.
Oh, I forgot about the "clear advantage" I mentioned. If the citizens of the anarchy had distinct military advantages over outside rulers or wanna-be tyrants, it would have a much higher chance of success. If people had the ability to rise up and destroy anyone trying to claim total ownership of a owner-less society, it would have a higher chance of success. If the citizens are not armed and organized, however, anyone can rise up under the pretense of "defense".