anarchism is not about abolishing government, it's about abolishing the state, which is a different thing. the idea is that you replace the state with democracy; you replace the state with a government.

definitions of the state are inexact. it's a similar idea to what the right calls the "deep state"; anarchists just call that the state. abolishing the state means abolishing private property (in the means of production). it means collectivizing production. it means democratizing everything. it means taking power away from an unelected upper class and giving it to the masses of people, instead.

the system of government sometimes called "market libertarianism" is not anarchism in any coherent sense, but is in truth fascism in disguise, as you need a totalitarian state to apply the market theory, and the aims and goals of such a system very quickly fall off into social darwinism and then nazism, if left to develop unchecked.