No, it does not mean this. There are at least two reasons why it does not.
First, there much of the government that we experience as Americans is state or local government. The police who protect our homes are local government officials. Our schools are run by a combination of state and local officials. These levels of government will be relatively unaffected by the shutdown. This means that the parts of government that we interact with most often will not disappear.
Second, the shutdown does not mean that all aspects of government will shut down. Instead, it is mainly going to be the parts of government that are designated as “nonessential” that will close down. The military, for example, will still provide national defense. Air traffic controllers will still make sure that airplanes do not crash into one another. Prison guards will still ensure that federal prisoners do not leave prison. All of these essential operations will still be going.
Thus, even though we will have a shutdown of things like national parks, we will still have a government performing its most vital functions.