Don't get me wrong; as an Alaskan, I'd love to see this go in, but it's not going to happen.
First, the seas of the Bering Strait makes the English Channel look like a backyard swimming pool, especially in the winter months - that is, most of the year. Boats simply don't go there. Good luck operating any kind of floating construction equipment with colder than -100F wind chill, ice bergs, and 1-2 knot constant water currents.
Don't underestimate the difficulty in putting a rail system across the land in those areas either. Between Fairbanks and Nome is literally five hundred miles of permafrost, swamps, silt, fault lines, rivers, lakes, mountains, and simultaneous combinations of the above. There isn't any infrastructure or construction-friendly ground here. Even with ten thousand people living in Nome we can't even build a dirt road out there. I can't imagine the other side of the Strait being any better.
And let's not forget the environmentalists. If we can't drill ANWR, then this rail is out of the question.
To call this a pipe dream would be an understatement, especially for a measly $65 billion.