The important thing to keep in mind is, that we live in anarchy right now. This, what we have, is exactly what happens when you leave a bunch of homosapiens on a planet to live in chaos.
So, if we did do away with all the police, it would really just be temporary. However, I do think you could do away with military for the most part, and the world would not be affected so much.
If all the police vanished, and it took a month to fix it, I think that for the most part life would go on more or less the same as it is now in the beginning. But eventually the word would get out, and problems would start.
For me, having no police would actually put me off of committing crime. At the moment, if you have 50.00 bucks cash in your house, and I come steal it and get away with the money, and you don’t have proof, there is not much you can do, the police wont help you, and if you hurt me you are the bad guy. However, with no police, there could be any consequence for anything.
I think that you would find, that some parts of society would club together sooner than others, and you would get pockets of no law and vigilante law, until the network gets big again.