Let's say you're a 20 year old Spartan citizen and you're sick of it and its culture. Maybe you've been disgraced in front of some people, or you want to try being a merchant in Thrace or a wandering philosopher, or you're just plain sick of Sparta and its culture. You don't like life in the barracks and you want more freedom in life.
Could you get up and leave? Would your wife and children (I know most Spartan men didn't marry until later and might not have any legitimate kids, but let's say he might) have to leave with you?
It seems like Sparta is like a police state. Could you get out of Sparta and move somewhere else?
I'm going to address this question in two parts: one, could they, and two, would they want to?
First, could they? If you were a Spartan citizen and wanted to leave, nobody would stop you. However, being a citizen (or perhaps more appropriately, retaining citizenship) meant you were expected to remain in the city unless given permission (or more accurately, ordered) to leave because you were also expected to be available for levy. I am not aware of any ancient sources that go into great detail as to how actively this was enforced of course. As to whether you could take your wife and children with you, well, that's a bit more complex. Your example of a 20 year old suggests someone that would have very young children, so I don't see any reason why you couldn't leave the city with them. If they were a bit older they would be enrolled in the agoge so that would add another layer of complication.
The second question though is, would you want to? I would say without a doubt that the answer is no, for both you and your wife. Spartans had no movable property and wealth; their power and wealth was derived from their military prowess and control over slave estates, not movable wealth. If a Spartan was to leave, by their own accord or not, they would not really have anything with which to begin a new life except for the former. Indeed, this was the case for most exiled Spartans, as they tended to find themselves under the employ of foreign polities in a military capacity as soldiers/generals. So, really, you'd just be continuing the same life you had in Sparta, except outside your own city. And therein lies the big issue with this: loyalty and duty to your city state was already a big deal in the polises of Ancient Greece, and Sparta took this to an almost fanatical level. To abandon that duty was shameful at best, and I have a hard time seeing a plausible reason one would abandon that duty in a Spartan's case.