Transnistria, a gap between the current and original course of the Dniester river, claimed by neither of its neighbours (Moldova, to the west, claims the original, western course to be the true border, while Ukraine, to the east, cliams the eastern course to be so), has declared itself a state, undergoing a war of independence ending in a treaty to which Moldova, Ukraine and Russia were party, is a great example of people taking the chance to establish a country. Sealand is a much more harmless and playful example, a Maunsell Fort* claimed by the Bates family.
*A number of offshore platforms built off of Anglia and in the Thames and Mersey estuaries during WWII. The most well known are probably Sealand (HM Fort Roughs) and Redsands (U6), which has been used for filming Dr Who and Danger Man, IIRC, and was home to Radio 390, one of several pirate stations located on the forts.