I hope it's OK to post to an old thread, but I think I have something to add.
Dmitry Zinoviev has created his excellent
Russian, CIS and Baltic Railway Map, showing the ex-USSR railroad lines, complete with styling and color to show various features of them.
First off, there's a rather obvious gauge compatibility problem. The US and Canada use standard gauge, 4 ft 8 1/2 in / 1435 mm, while the ex-USSR, Finland, and Mongolia use Russian gauge, 4 ft 11 7/8 in / 1520 mm (standard gauge is originally 5 ft on the outside, Russian gauge 5 ft on the inside). However, that incompatibility can be handled in ways that it is handled at several other standard/Russian meeting points. Shipping containers can easily be moved from a standard-gauge flatcar to a Russian-gauge one, and vice versa.
And then there is the sheer length of new rail line that will have to be built -- on the Russian as well as on the North American side.
The closest that the Russian railroads approach the Bering Straits is Yagodny on the Amur River, about 100 mi / 160 km northeast by road and rail of Komsomolsk-on-Amur (Komsomol'sk-na-Amure; pop. 280,000). It's about 1200 mi / 2000 km to the next sizable town, Magadan (pop. 99,000), about 1200 mi / 2000 km more to the farthest-east Russian town, Anadyr (pop. 11,000), and about 600 mi / 1000 km more to Uelen (pop. 500) at Cape Dezhnev, on the Russian side of the Bering Straits.
This yields a grand total of 3000 mi / 5000 km, though I estimated this distance by trying to follow the coast as much as was reasonable. Much of the Russian Far East is rather mountainous, which is why I avoided a more direct route. The great-circle distance is 2000 mi / 3000 km, but that crosses the Sea of Okhotsk and the aforementioned mountains.
However, Komsomolsk-on-Amur is on the Baikal-Amur Mainline, which runs parallel and typically 400 mi / 700 km north of the Trans-Siberian Railroad. It splits off at Tayshet, halfway between Krasnoyarsk and Bratsk. It is electric and double-track to Taksimo, about 200 mi / 300 km east of Lake Baikal, and diesel and single-track all the rest of the way to the coast.
From Komsomolsk-on-Amur to Khabarovsk (pop. 580,000) up the Amur River is about 200 mi / 300 km southward; Khabarovsk is on the Trans-Siberian Railroad, an electric double-track line from Moscow to Vladivostok, also on the coast.
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On the American side, there is a similar long stretch of track to be built to connect to suitable existing railroad lines.
The Alaska Railroad runs Seward - Anchorage - Fairbanks, and thus would be unsuitable.
The closest town connected to the railroad lines of the Lower 48 and Canada is Fort Nelson, BC (pop. 5000). An Alaska - Yukon - BC line running to there would be about 1500 mi / 2500 km long.
Canadian National runs a line from Prince George, BC (pop. 71,000) to there, though strictly speaking, it ends about 4 mi / 7 km to the south in Muskwa. With Google Maps, one can see a yard in the west side of that place.
However, with increased traffic, it may be better to build a straighter line, and one that avoids the mountains near Prince George. A Fort Nelson - Edmonton (pop. 1,000,000) line would be 500 mi / 800 km long, and would avoid the Rocky Mountains. The existing Fort Nelson - Prince George - Edmonton route is about 50% longer.
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The total Edmonton - Khabarovsk rail distance is thus about 5300 mi / 8500 km -- almost as long as the Trans-Siberian Railroad.