I've long been fascinated by micronations. It's the geek obsession with taking fundamental building blocks and trying to reinvent them. Following their histories also reminds me how disastrous applying that engineering instinct to social problems can be!
The first of the modern micronations was Sealand, and it's still up and running. Started by a 60's pirate radio broadcaster on a WWII sea fort, it's gone through everything from firearms arrests to kidnapping and helicopter assaults. It even has a government in exile!
The Republic of Indian Stream was an accidental state on the US/Canadian border in the 1800's. An ambiguous border treaty led first to both countries trying to tax the inhabitants of the 300 square mile area, and in protest they declared their independence. The situation lingered on until the inhabitants invaded Canada to free one of their neighbors imprisoned over a hardware store debt. At that point the ambassadors decided an international incident over money owed to a shopkeeper was too embarrassing and negotiated a border agreement ending the republic's short life.
The Territory of Poyais never actually existed, but Gregor MacGregor still managed to raise hundreds of thousands of pounds in the 1820's to settle it. The unlucky colonists who made it to the supposed tropical paradise found themselves stranded in hostile jungle. One committed suicide, the rest were eventually rescued by a chance encounter with a Honduran ship after their original vessel was swept away in a storm. When the survivors made it back to London and spread news of the scam, MacGregor fled to Paris to try it all over again!
Probably the only place out of these I'd actually want to live isn't a real micronation since it doesn't claim to be its own country, but Spiral Island deserves a mention. It's a floating island built on a raft made of 250,000 recycled plastic bottles, complete with a two storey house, beaches and even palm trees. Unfortunately the first version succumbed to a hurricane, but Richie Sowa, the artist behind it, rebuilt it with even more amenities.