Story, in his second season, suffered the injury while batting in Tuesday night's 10-4 victory over the Chicago Cubs.
Story has struggled this season, hitting .180 with six homers and 15 RBIs after a big rookie season in 2016 in which he set the National League record for home runs by a first-year shortstop. He also tied a major league rookie record by hitting 10 home runs in April.
He finished with 27 homers last season before going on the disabled list in August with a torn ligament in his left thumb.
Pat Valaika is starting at shortstop Thursday night against the Dodgers.
Manager Bud Black decided to activate Hoffman, one of the team's prized prospects, and give another few days of rest to Anderson, who tweaked his left knee in his last start. Anderson is scheduled to start Saturday.