The wikipedia article on "market failure" seems unusually balanced. Often I read of market failure as an established fact, like man-made global warming.
In practice, I see market failure used as an excuse for all sorts of government action. In most cases, I think there is more a failure of imagination than of the market. The market will usually find a way to satisfy human needs. Other non-aggressive means, such as charity, also can work.
The classic example is the lighthouse. See the story here: http://www.independent.org/aboutus/lighthouse.asp