It already does.
Anything that isn't directed by a government agency or a central corporate cabal already exists in a form of anarchy. People do what needs done. They talk with each other to decide what needs done. They live their lives, and they let a little anarchy in. Governments and corporations just hang around the edges, pretending to be necessary.
"But the market provides ..." The market doesn't provide shit. People provide. And they do so because of a common mass-delusion involving computer bits representing colored pieces of paper representing the shiny rocks that used to sit in a now empty vault....? And if tomorrow there were no more bits or papers or shiny rocks, then they'd probably get up and do it anyway - if what they did was useful - because it was useful and someone was counting on them and because they were actually human beings and not just a factor in some vicious economic equation.
The current classic is
http://theanarchistlibrary.org/library/peter-gelderloos-anarchy-works
it is long but well indexed. If you have a "yeah, but..." Peter has probably already answered it with a practical example.
I'll add this, just because i personally like it:
http://theanarchistlibrary.org/library/margaret-killjoy-take-what-you-need-and-compost-the-rest-an-introduction-to-post-civilized-theo
especially the section:: If We Had Our Way
"What does a city look like if it’s not a city anymore?"
you probably won't find it reassuring. But I'd point out that cities are more figment than fact, mostly villages that grew until they ran into each other and melded along the borders. Some shared infrastructure sure, but even now surprisingly different socially and culturally, even a few blocks away.
There are probably others, that'll do for a start.
Good luck.
(ps: plant edibles, you're probably gonna need them.)
