Can a 16 year old move out?

It depends. You could move out with your parents permission, but if they won't let you you have to go through the emancipation process. In order for the State you live in (assuming this is question is coming from a US teen), there has to be a very good reason to be emancipated (ie, you are being abused) and you have to prove that living on your own would be more beneficial to you than moving in with a relative.