I'm planning a trip and one of my main points is not to fly unless absolutely necessary. My biggest problem is how to cross from Asia to America without a plane. If it's absolutely impossible to cross the Bering Strait without flying, which is the shortest route? Maybe a small airplane?
Crossing Bering Strait on a boat does seem possible but it's going to be difficult practically. Adam Katz put together a bunch of notes on how to do this. From the looks of it, it appears you might have to pay your way through it on commercial fishing vessels which operate in the area.
According to this resource, navigating Bering Strait is hard because of weather/geographical conditions and the legality of getting permission to leave Russia. From the linked source:
It is recommended to cross in a seaworthy vessel capable of handling intense storms. It is possible to cross the narrowest part of the Bering Strait in a smaller boat such as a kayak during a calm period of weather, however a support boat would be recommended. The problem, however, is the possibility of having the boats confiscated upon reaching shore.
and (assume there are similar problem the other way)
How do you cross the Bering Strait legally?
This is very difficult. Not only is it necessary to arrive in Russia in an official port of call, but it is also necessary to depart from an official port of call. We have not heard of any adventurers who have received permission to arrive or depart from the remote shoreline of Russia.
A discussion on Lonely Planet's Thorn Tree forums mentions previous attempts and further information. A pertinent point is while it may be possible to cross, going on from there onwards to rest of Alaska/America has its own problems too.
It should be noted that Nome is not a usual port of entry into the USA, and it could be a problem as there is only a part time Immigration and Customs agent. In addition, from Nome to the rest of Alaska would then require an expensive flight to either Anchorage or Fairbanks. Also from Provideniya there are only occasional flights to Anadyr, the Chukotka capital that is considered one of Russia's less desirable destinations.