Move to beautiful, sunny Bir Tawil!
Are you thinking of starting your own country? It's not that crazy an idea. Citizens of existing states have already successfully declared de facto indepedence and carved out tiny, mostly unrecognized sovereignties for themselves.
So how to start? You could take over an abandoned sea platform, but these days you're just as likely to get charged with terrorism for seceding as you are to build a new state. Declaring your own backyard an independent political entity is also an option, but it's a lazy one, and you'll still be bound by your homeowners association bylaws no matter how many medals you stick on your presidential military uniform.
No, what you'll need is a piece of real, solid land. You might think every patch of soil on Earth has been claimed by one country or another at this point. Hell, some countries can't get enough land - there are currently hundreds of pending international territorial disputes. But you'd be wrong.
The land nobody wants
Bir Tawil is a 2,000 square kilometer territory wedged between Egypt and Sudan. Its name means "deep well" in Arabic, presumably because there's a deep water well somewhere in the area. This patch of rock and sand east of the Nile is landlocked and has no natural resources or permanent settlements, and it is unclaimed by both Egypt and Sudan - each country's government insists that the other is Bir Tawil's legal administrator. Lack of sea access or resources has never stopped a country from claiming land, though. Why don't Egypt or Sudan claim Bir Tawil?
In 1896, the British-controlled Khedivate of Egypt launched a joint Anglo-Egyptian military expedition to retake control of the Sudan, which had revolted. By 1899 the reconquest of Sudan was complete, and Britain formed the condominium of Anglo-Egyptian Sudan. The situation called for the redefining of the Egyptian-Sudanese border, so Britain drew up a new map that cut the two in half straight across the northern 22nd parallel. In 1902, however, the British authorities changed their minds and redrew the eastern part of the Egyptian-Sudanese boundary, giving the small territory of Bir Tawil, just south of the 22nd parallel, to Egypt, and the much larger 6,500 square kilometer Hala'ib Triangle to the north of the 22nd to Sudan. The reasons for this shift were pretty logical: the residents of the region of Hala'ib identified as Sudanese, while Bir Tawil was commonly used as grazing land for the Egyptian Ababda tribe.
Unforunately, this shift eventually resulted in a territorial dispute. The now independent Republic of Egypt insists that it is the legal owner of the Hala'ib Triangle according to the 1899 border (and is currently the de facto administrator of both Hala'ib and Bir Tawil) while the Republic of the Sudan rejects the 1899 border in favor of the redrawn 1902 border, which gives it control of Hala'ib. The strange quirk of this dispute is that, at the same time, both countries deny ownership of Bir Tawil. There's no legal basis for either country to claim both territories, and while Hala'ib is much larger than Bir Tawil and borders the Red Sea, Bir Tawil is small, landlocked and pretty much has nothing going for it.
Establishing your new country
If you want to establish your new sovereign state of Bir Tawil, you may face a few obstacles. You'll first have to somehow deal with the fact that Egypt currently occupies the region and that, even though the Egyptians would rather not have it themselves, they'd probably insist that Sudan should have it in order to validate the 1899 border between the two countries. Meaning they probably wouldn't be willing to hand it over to some random punks.
Even if the Egyptians do agree to give up control of Bir Tawil to you and your friends, though, it probably wouldn't matter. Bir Tawil has no natural resources (unless sand and rocks count) and it's landlocked between two states, so any imported resources would have to go through either Egypt or Sudan, with all the duties and bureaucratic holdups that would involve. Water and electricity would also probably be an issue. Unless you're a herder from around Aswan, Bir Tawil has little to offer.
You'll also face some competition. Despite being practically the definition of the middle of nowhere, several micronations have already declared ownership of Bir Tawil. The prospective governments of Bir Tawil don't have any physical presence in the region, however, making their claims irrelevant. By sending over your own force and establishing a presence and a permanent settlement in Bir Tawil, you can get a solid hold on your own claim. Considering all the downsides, though, you might be better off tying together a bunch of plastic bottles and building a floating sea nation.
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Related sites
http://googlesightseeing.com/2009/11/bir-tawil-triangle
Satellite maps of Bir Tawil and the Hala'ib Triangle and more background on the Egypt-Sudan border dispute.
http://kingdomofbirtawil.blogspot.com
These guys represent the "Kingdom of Bir Tawil", one of several prospective micronations that have claimed the unwanted territory. You'd think a sovereign state would be able to afford a better site than a crappy Blogspot page. They stopped updating three years ago, so their claim has probably lapsed by now.
http://micronations.wikia.com/wiki/Micronational_claims_of_Bir_Tawil
A whole lot of other "micronations" that claim Bir Tawil. Most of them have silly names.