The fact that Chukotka is sparsely populated wouldn't be such an obstacle in itself. It could even be advantageous = no NIMBYs preventing the construction of the railway. The intermediate steps are manageable - after all the Transsiberian Railway was built in the wilderness in the 1890s and early 1900s with much more primitive technology than is available today.

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.