The Colorado Rockies of Major League Baseball (MLB) selected Story as a shortstop in the first round, with the 45th overall selection, of the 2011 MLB draft. Story signed with the Rockies, receiving a $915,000 signing bonus, rather than attending LSU. After he signed, he played for the Casper Ghosts of the Rookie-level Pioneer League, where he batted .268. In 2012, Story played for the Asheville Tourists of the Class A South Atlantic League (SAL), and was named a SAL Postseason All-Star after batting .277 with 18 home runs, 63 runs batted in (RBIs), and 43 doubles, which led all SAL players. Prior to the 2013 season, Story was ranked as the 99th-best prospect in baseball by MLB.com. Story played for the Modesto Nuts of the Class A-Advanced California League in 2013, and he struggled, recording a .233 batting average with 12 home runs and 65 RBIs. In 2014, after spending a short time with the Class A Short Season Tri-City Dust Devils and excelling in Modesto, with a .322 batting average and 20 stolen bases, the Rockies promoted Story to the Tulsa Drillers of the Class AA Texas League in June. He struggled with Tulsa, batting .205.