Bring blackberries, sugar, and ½ cup water to a boil in a medium saucepan, stirring until sugar is dissolved. Reduce heat so that liquid is at a simmer and cook, stirring occasionally, until fruit is very soft, 20–25 minutes. Let blackberry syrup cool, then strain through a fine-mesh sieve into a large pitcher, pressing on fruit to extract as much liquid as you possibly can; discard solids.