There are many things which the private sphere cannot do, and other things which it could do in theory but has utterly failed at in practice. The obvious things it cannot do in theory or practice are set laws - the rules we all must live by whether we want to or not. Things like you don't get to murder other people on a whim. That is no something the private sector can enforce, and if you argue that society could simply enforce that through the majority exiling people who commit those sort of crimes - what you are describing is government, in its most primitive form. The private sector cannot handle this - not even in theory.
As for things the private sector could handle in theory, but has utterly failed at in practice, health care is an obvious one. If you compare the places where the government handles health care as a private good well to places where health care is handled by the private sector, the private sector has not outperformed the government - anywhere.
It really boils down to this: not everything can be voluntary in a society. Some rules have to be set, and enforced by everyone with no opt out. We can debate endlessly about exactly which set of rules should be included in that, but there's a minimal set (don't kill, don't steal, etc) that almost everyone can agree on. The process of agreeing upon those rules and enforcing them is, by definition, government. And having those rules exist and enforced is, by definition, civilization.