It's not just the technical challenges of building a bridge or a tunnel in pretty harsh conditions. It's also that on the russian side there is a huge area that has almost no human settlements and no infrastructure. You'd have to build a train line through an area of nothing. That also means it's difficult to imagine intermediate steps (e.g. first building half the rail line to even get there).
It should be noted that just because there's no major settlements in a vast area doesn't mean there's nothing there. The Alaska/rest of North America rail connection is constantly being proposed on the basis of being able to hook up mineral extraction operations in the region, although these are marginal enough that the proposals keep falling through. Similarly, the Yakutsk-Magadan region once hosted several industrial concerns along the highway there, but the fall of the Soviet Union caused those production centers to lose viability and collapse.
But the terrain between Yakutsk / Nizhny Bestyakh (the closest rail terminus) is pretty bad, bad enough that the construction does not make economical sense at all. The soil moves during the thaw/freeze cycle etc.