I would like to do this myself also, Ive been looking up various information and the only transportation Ive been able to find is flights. I would like to get a vehicle across though, and Ive heard there are helicopter ferries that can transport vehicles. Im not sure whether I want to try to drive across the ice or not.
There are several barriers to crossing the Bering Strait (although I, too, would love to try it):
1) There are NO ROADS to the Bering Strait (on either the Alaskan or Siberian sides) so driving there in a car (as a couple of people mentioned) is out of the question.
2) The Bering Strait almost never freezes over completely, so crossing by foot in the winter is well-nigh impossible.
3) Even if you cross over from Alaska to Russia, the Russians will simply deport you. Unless you have a special visa to visit this militarily-sensitive part of Russia, which takes over a year to obtain, according to the information I've seen, and requires you to submit a detailed itinerary, and use a tour company.
I'm not trying to discourage anyone from trying to cross the Bering Strait, but it will take a lot of time, money, planning and stamina. It's not like driving out to the shopping mall.
I live in Alaska and am also familar with Unalakeet as a previous poster mentioned it. My Grandfather was Station Manager for Alaska Airline in the 1970s. You had better make sure all your permits for both sides are in order. Also where is your fixed destination in Alaska? You mentioned Nome? You should make sure your hotel/motel reservations are already booked or your going to be hanging out in a bar and taking showers at a laundromat and no where to sleep. Also plane reservations from Nome to at least Anchorage should be firmly in place. Flights are going to be booked up. You need to be absolutly certain that you have enough money for the Alaska side of the border as Alaska is quite expensive. If you try to walk you take the chance of the ice litterly breaking under your feet and you drowning. Your litterly going to be out in the middle of nowhere. Thier are not a lot of facilities on either side of the border at that point. How do you plan to get to the Russian side of the border? Then keep in mind that fisherman dont just randomly go back and forth. Thier crossing an international border so they have to have thier passports and permits and visas in order. I would suggest you do a ton of research and really think about this. Its very critical as well that you have the proper cold weather gear. Febuary in Russia and Alaska is not the place to be trudging around unprepared. Im very concerned for you and hope if you decide to do this. That you really really think about it. I am an international traveler but this is serious business and Death is not funny. Good luck,