I am pretty sure when I read the rules they said you can use one of your turns to claim a story from the table or play one from your hand (if you meet the requirements). I feel like that is what the rules said, or how I read them.
Well, that's not what my copy of the rules says. It says you can claim a story from the table or from another player (if you can exceed their count of the required story icon).
The setup directions are also pretty clear: Stories don't go in the draw pile. So they'll never be in players' hands.