What do we mean by the term “market” and what do we mean by “government”?
As a first approximation, I am going to define the market as the sphere of activity where everything is voluntary. All trade is voluntary. All transfers of income and wealth among people are voluntary. I am going to define government as the sphere of activity where everything is coercive. Government puts limits on private behavior and enforces those limits by fines, taxes, imprisonment and even death. Also government may force private citizens to do what they otherwise would not have done – again, under threat of force.
So everything that is voluntary is a market activity. Everything that is coercive is a government activity.
The very idea of “failure” implies that we have an idea of what success looks like. In the economic realm, that idea is very well defined. An ideal economic arrangement is one in which people have exhausted all opportunities for mutually beneficial exchange. Once they have done that, there is no way to make one person better off without making someone else worse off. Such a condition is called Pareto Optimality.