It's you who logically failed here. You pay some bad guys and they defend you from other bad guys out there. Because of the economy of scale, you actually end up paying much less than you would have to pay without a state behind you. So your imagination wasn't actually deceiving you...
You clearly have no idea how much value is taken from you in the multitude of ways that the government takes value from you, and greatly overestimate the cost of defending yourself in a free society.
I have already answered almost the same assertion in my previous post... If you don't believe me, Google is your friend, go search for the Laffer Curve. I'm afraid that "in a free society" you will often have to give everything you possess and sometimes even your life. I have a strong feeling that anarchy proponents here have never seen what real life may look like...