What your proposing sort of exists at the moment - some countries (the US and the UK for example) have mechanisms in place to allow you to renounce your citizenship. It's not quite what you are proposing since you usually need to have an alternative citizenship, so hypothetically let's say we remove that requirement.
You can now go and start your own society elsewhere - finding some land that has no recognized claim against it is difficult though. But for the sake of argument let's say you discover a brand new island that's located outside of any existing territorial waters and has yet to be claimed. You could go there, plant a flag or whatever and issue a proclimation declaring yourself King rus9384 of Rustopia and have whatever conventions, laws, rules, and societal structure you wanted (essentially this is what lead to the formation of America)
Congratulations! You are the new ruler of your very own micronation. The difficult bit comes in getting other governments to recognize Rustopia as a legitimate state. Assuming your new homeland can support sufficient agriculture or hunter/gatherer actions to support you for your remaining lifespan it probably won't make much difference whether they do or don't.
On the other hand if you actually want to interact with other nations, maybe engage in some trade, or even go on holiday to another country you might have problems.
There are of course other problems as well, which essentially mean a society cannot exist in true isolation - since we've given that same freedom to renounce to everybody else maybe someone else (let's call him Big Bob) has the same idea as you and decides to establish The Grand Empire of Bob on "your" island.
After a long, harrowing, 30 mins of kicking each other in the nuts you both decide that it's going to be better for everyone if you agree to share the island and the "Ow, my testicles really hurt!" accords are signed, stipulating which bits of the island are Rustopia and which are The Grand Empire of Bob.
Now you, as a citizen of Rustopia have a duty placed upon you by virtue of your citizenship. Namely not to go to any of the areas designated as Grand Empire of Bob territory, of course since there is only the two of you there's no police, no courts, and no judges to stop you walking into Bob's territory. But if you do that he's just going to start kicking you in the nuts again.
Of course if you find the constraints of the accords too restrictive on you then you are of course free to renounce your Rustopian citizenship and go somewhere else. But since habitable locations on Earth are finite (and most of them are already part of one nation or another) you aren't going to be able to do that many times before you run out of places to go.
So in short the finite nature of Earth's resources and the current high usage levels thereof (especially habitable land) means that if you want to be able to settle down somewhere and live your life without fear of getting kicked in the nuts repeatedly by someone of another society (other unpleasant consequences of a dispute are available) you're going to have to accept some limitations placed upon you that come from other societies.
Should humanity find itself in a position to feasibly expand off-world then I expect you'll see some new societies forming then, but unless they can establish themselves to be truly self-sufficient then similar inter-societal compromises will have to be formed.
Earth is essentially full, so the notion is currently a non-starter.
PS: Not that Earth's fullness has stopped people trying - there have been dozens of attempts to start new countries over the last 150-200 years alone. "The Kingdom of North Sudan" is a recent example - Bir Tawil , a strip of officially unclaimed land (terra nullius) that a man named Jeremiah Heaton claimed as "The Kingdom of North Sudan" in 2014. Unfortunately for the aspiring monarch no other government recognized his claim and that is where the whole thing just sort of fizzled as it has done for so many Micronations over the decades.