Technically doable and I think there could be economic justification for oil pipeline
The technical challenges are formidable but doable. The Bering has two islands to each underground tunnel or bridge would be about 22 miles long.
The Alaska oil pipeline is running at less than half capacity. The US does need a lot of oil. Making an oil pipeline and high speed fiber connection could make sense. Russia is the world’s largest producer of oil.
A full Trans-Alaska Pipeline means a maximum capacity of just over 2.1 million barrels a day. Alaska oil supply (production) will decrease from an average of 600,000 barrels per day in 2010 to 520,000 barrels per day for 2012.
Does not make sense to connect people
There are no people nearby the Bering Strait to connect.
It is 1,327 miles miles from Anchorage to Vancouver.
Crossing the Bering Strait is a challenge but it is doable.
It is still over 3000 miles to connect anything on the Russia-Asia side with Alaska and about 5000 miles to go around to get to Vancouver and Seattle.
Metro London has 13 million people. Paris has 2 million but there are about 55-60 million people in both France and England. Lyon (south of France to London – 570 miles).
Competition from Shipping
Container ships are as efficient and even more efficient than rail. Rail can be faster and efficient but not if you are sending things thousands of miles farther than they need to go.
BTUs per ton mile: barge 433; railroad 696; oil pipeline 1,850; truck 2,400 (USDOT Maritime Administration.)