Friday afternoons at the Chop-Chop Barbershop in central Moscow used to be busy, but at the beginning of a recent weekend, only one of the four chairs was occupied. “We would usually be full right now, but about half of our customers have gone,” said the manager, a woman named Olya. Many of the clients — along with half of the barbers, too — have fled Russia to avoid President Vladimir V. Putin’s campaign to mobilize hundreds of thousands of men for the flagging military campaign in Ukraine. Many men have been staying off the streets out of fear of...