CHERNIVTSI, Ukraine—One day last March, Iryna Shabanova and 18 members of her family started running out of food. They were hiding near the burning Ukrainian city of Mariupol — which had been under Russian attack since the start of the war — and trying to survive the siege and the explosions they heard every few minutes. In desperation, they had taken to drinking water from ancient cast-iron heating radiators, and even that had run out. They had to get water. They had to eat. Shabanova’s husband and another man started walking to the city’s downtown to find help. They never...