The emphasis is more on "Free" than on "State".
From m-w.com, the first definition of "State" that's at all relevant. We could argue semantics of a state, but I don't think it's necessary:
5a : a politically organized body of people usually occupying a definite territory; especially : one that is sovereign
Certainly "Government" != "State" -- They are two quite distinct concepts. A Government is at least a property of, and at best is a tool of a State.
Based on the definition above, you need at least two things to be a state:
1) A body of people, and
2) A government (the property of political organization).
The government is the political organization of those people.
The plain text of the Law (the Second Amendment) reserves the right to bear arms specifically to the People, and not to the State, or the Congress, or anyone else (including "the Militia").
I believe that, regardless of whether it's more trouble than it's worth in today's civilization, the *intent* of the Second Amendment was to provide the possibility of organizing a Militia. The Militia is needed to protect the Freedom of the State (which is primarily comprised of people, not a government). The thing that the Freedom of the State needs protection from is The Government.
She ate the spider to catch the fly...