VOLGOGRAD, Russia—Sergei Fetisov, a 23-year-old welder, signed on for one of the most ambitious projects in Vladimir Putin's Russia: rebuilding the remains of the once-mighty Soviet Red Army.

A cornerstone of that effort was the creation of special combat-ready units staffed entirely by professional soldiers, not conscripts. Mr. Fetisov volunteered to be one of them. He enlisted for a renewable three-year stint, enticed by higher pay and the chance to learn new skills.

One...