The Chukotka autonomous district is notable as being the closest point that both Eurasia and Russia gets to North America and the United States. In fact on a clear day you can even see across the Bering Sea which separates them into Alaska. While Chukotka is massive at about 285,000 square miles, it only has a population of 55,000. With fewer than 400 miles of road and no railroad infrastructure; the population is mostly employed in mining and subsistence hunting.
Being this close to the US provides some interesting ways of getting into Chukotka. It's even possible to swim or even walk across (when the water freezes) but doing so will definitely get you into trouble with local border guards. The easiest way of getting in legally would be by either boat or plane, as long as they stop at one of the established points of entry. Remember that a valid passport, Russian entry visa, and Chukotka border zone entry permission are required.
Chukotka is listed among border regions with special permission requirements for entry of Russians and foreigners alike, so please contact a tour agency with knowledge of this area to obtain a 'rasporyazheniye' invitation document. Passengers of cruise liners are allowed to land on Chukotka and stay for 72 hours without visa and special permission.
Although highly impractical, it is also possible to swim from the United States of America to Chukotka across the Bering Strait. To date only the legendary Lynne Cox has chosen this route, swimming 3 kilometers/2 miles from Little Diomede in Alaska to Big Diomede of the then Soviet Union. Reportedly, she requested a "babushka" for when she arrived, thinking she could take home a souvenir scarf. But babushka in Russian (as opposed to Polish) means "old lady"—so the Soviets made a babushka available at the site of her arrival, trained in medical care to revive the freezing traveler to health!
Classic Chukchi literature of Yuri Rytkheu.