The main problem I see with anarchy is that it's an inherently unstable state. If you ever have it, it won't last. If you don't have a government, a gang of thugs will take over and become a de facto government.
The situation between state governments right now is effectively anarchy, since they have no 'super government' to moderate their disputes or interdict when they use force against each other. This seems to be a fairly stable state with 'one world government' folks being considered the nutters.
I don't think it's completely implausible to imagine that it could be possible for people to interact with each other the way governments do today. Most people, just like most governments, just want to go on about their business and peaceably trade and interact with others. Some want to be bullies or takeover the world, but various affiliations between the more honest ones hold the more aggressive ones in check.
Though I can't imagine how you could make it work, I don't think it's so immediately obvious that there might not be some way to make it work stably. Everyone has a mutual interest in peaceful cooperation and everyone has a mutual interest in no group (at least that doesn't include them) getting too terribly strong.
The biggest problem is that even if you can imagine a stable anarchic human civilization, can you imagine any conceivable way to get to such a civilization, other than super-powerful aliens plucking it down?
Even if you don't think it would necessarily be stable over the very long term, governments really aren't either. It may well be that no human civilization is stable over the long-term, so this isn't specifically a disadvantage of an anarchic society. It may also be that governments inevitably become more totalitarian until they too either degenerate into horror and misery or are replaced.