If you own land anywhere there is nothing to stop you declaring independence. However, doing and keeping are two seperate questions. Declaring independence from any legitimate government will probably earn you a spell in jail if you try to fight recolonisation. The truth of the matter is that no-one will recognise independence unless you give them a reason to do so. Bosnia and Croatia have done so, and were recognised because their independence solved a problem. Gibraltar would also fall into that category, because, as an independent Gibraltar could not be invaded by another country, given the ties they have with the UK, and the assurances they would seek. However, declaring independence on land that is solely claimed by one party, or on disputed land that would be fought for is a different question. The only people who would grant nation status to anyone in those areas would be enemies of the current ruling class. We need only to look at Taleban Afghanistan to see that. Pakistan, Saudi Arabia and the UAE gave it recognition because it served a purpose. Or take pre Nixon China, which was the island we now call Taiwan. If someone in, say, Birmingham, declared independence, assuming it was permitted and not stopped by the rest of the land surrounding it, they would be forced to deal with the rest of the UK economically and militarily, unless they had Nukes. All the rest of Britain would need to do would be to block the Birmingham House's access to trade, and they would be forced to concede to any demands made.
The only way to declare independence is to (to quote Roosevelt) "Talk softly, and wave a big stick"
However, if you want diplomatic status, give me a call, and I will see what I can do before they stop me!