Did you know that there is a part of the continental United States that it not actually connected to the rest of the continental United States? It's called Point Roberts and it consists of the southern part of a peninsula that sticks down off of southern Vancouver just below the 45th parallel, which marks the border between the U.S. and Canada. I've been there. There is an actual border crossing manned by actual border crossing guards whose salaries are paid for by your (if you're an American) taxpayer dollars. These guards defend the 2,739 residents of Point Roberts from invasion by Canadian militants. They don't do a very good job. I visited the place in 2009 and was able to walk right past the checkpoint from Canada into the United States unchallenged. It was not until I tried to get back into Canada from whence I had come that I was challenged by a guard on the Canadian side.
This bit of geopolitical weirdness is not unique. There is, according to Google maps, a tiny bit of Austria that juts out into Germany. This piece of Austria is technically contiguous with the rest of the country, but the place where it connects is only about 150 feet wide, and there are no road that go through it. To get from this part of Austria to the rest of the country without hiking you have to go through Germany.
But the weirdest example of bits of country in random places has to be Oman. There is an enclave of Oman called Madha that is completely encircled by Fujaira, one of the United Arab Emerates. And completely encircled by Madha is the enclave of Nahwa, which belongs to the Emirate of Sharjah, the main part of which is next to Dubai. And as if that weren't bad enough, there is a third discontiguous part of Oman called Musandam at the northern tip of the Arabian peninsula.