Arguments for government and against private alternatives, both in general and in specific areas, are frequently based on the existence of market failure. In the first part of this essay I briefly sketch the idea of market failure and show that it provides problems for, hence arguments against, both private markets and the political alternatives. In the second part I discuss some of the imperfect ways in which private markets deal with such problems in the small, providing public goods or internalizing externalities within a framework of enforced property rights. The third part describes how that framework itself might be privately provided, examines difficulties that might arise due to forms of market failure within a system of private law and law enforcement, and considers how those problems might be dealt with. In the conclusion I argue that a society without a government, market in the large, could be stable under some but not all circumstances, and if stable could be expected to produce more attractive outcomes than a society with a government for reasons closely related to the reasons to expect the market in the small to work better than its political alternatives.