A nation is a group of people with a common culture, identity, and usually language. A state is a legal construct implying self-governance and monopoly of force.
"Nation" has become colloquially synonymous with "nation-state", which lead to your confusion. But the legal aspect is from the "state" part, not the "nation" part. There have been historical states (eg. the Holy Roman Empire, Hapsburg Austrian Empire, Ottoman empire) that were not nations, and there are present day nations (the grandparent mentioned some: Palestine, Kurdistan, Tibet, as well as Basques, Zulus, and others) that are not states. Some present day nation-states (eg. Germany & Italy) were nations since medieval times but did not achieve unified statehood until quite recently (1871). The U.S. started as a state (technically, 13 states and one federated government) but did not really become a nation until after the Civil War.
If you can convince other people to identify with you, you absolutely can found your own nation. It won't be a state, though, so while might have your own customs good luck on having your own laws.