Place onions in a bowl and cover with cold water; let sit until skins loosen, about 5 minutes. Peel and transfer to a 4-qt. saucepan along with 4 cups water; bring to a boil. Reduce heat to medium; cook until tender, 12–15 minutes. Add vinegar, sugar, and salt; cook until liquid is reduced to a thick syrup, about 1 hour. Stir in oil and oregano.