Methinks you have some baseless stereotypes as to what exactly anarchism as and how it functions. There are many, many groups today that practice anarchism, and they're quite stable. Noteably, Spain following Franco's uprising was almost completely run by anarchists (the CNT). They managed to organize the peasant population into a military force effective enough to stop the advance of the Spanish army ffs. People with no military experience, not even the ability to read, with nothing but the will to fight for their freedom, were able to stop the Loyalists dead in their tracks and hold line, working (yes, within the military as well) on the principle of anarchism. Sounds plenty organized to me. Most of the Communist forces that would later fight in the war were trained and equipped behind the shield that the anarchists and others provided, before they were suppressed by the Communists themselves.
Remeber, people supportive of monarchy said the same thing of democracy, that it would be too disorganized and could never function.
I'm not saying that its the best system to have, not for the time being anyway, but its a very unique and interesting alternative. Most people, myself included, are not really ready yet to accept such a radical departure from current political organization (e.g. democracy), but hey, its an option.