Originally from Kiev, Ukraine, Babchuk began playing hockey at the age of three. He began his training with the Sokil Kyiv junior hockey affiliate, a team which included fellow future Ukrainian NHLer Nikolai Zherdev. While participating in an overseas peewee tournament in Quebec, Canada, Babchuk's team found itself competing against a team from Elektrostal, Russia. The rival team's coach, Ravil Iskakhov, took note of both Babchuk and Zherdev, and invited the pair to further their development with the Elemash Elektrostal hockey club of the Russian Major League, to which they accepted together.