This entirely depends on the angle you want to write from. If you want to write about a young child's first experience at Yankee Stadium/Fenway Park/ Wrigley Field, then there is no need to know anything in depth about baseball outside of the average fan.
If you want to write a story where the MC is one of the players, you will need to learn about the strategy of the game. Focus on something specific. Maybe a pitcher, maybe a batter. Don't try to learn the whole game as it will overload the short story. Learn the game from that position's perspective. As a pitcher MC, you can focus strictly on pitch sequence and what type of ball get's thrown in what situation.
Learn the lingo. Hanging curve, a cement mixer, high heat, putting it in the ear, around the horn, chin music, cheese, gas, going yard, dinger. There are many slangs and terms used in baseball that will help provide an authentic feel.
Baseball is a romance story in itself. It has it's ups, downs, betrayals, heart break, tension, euphoria. It has it's games within the game (fake signs, faking movement behind the batter to make him think the pitch location is different from actual location, pick off attempts, challenging a batter vs pitching carefully to a batter). There is literally so much you can do within the realm of baseball.
You just have to narrow it down to a specific instance, a moment, an inning, a feeling and write about it. There is no timer in baseball, a moment can literally last a life time. You can follow a similar format to the movie "For Love of the Game" where the main story was about a single baseball game and in particular about a specific pitcher that used flash backs between innings, pitches, to tell the story.
For a short story, I would focus more on a critical moment of a game. The last inning, the game winning hit, the game saving close. Write about the emotions someone would go through. All the checked pitches, catcher visits. The thoughts about what pitch and where. Is the batter aggressive or does he take the first pitch? Does the pitcher throw sliders on a 1-2 count for the strike out? All of these little mind games adds up to what makes baseball so great.
Baseball is beautiful sport that forces teams to earn their win. There is no killing the clock, pinning the ball to a corner or a puck to the boards. You can't avoid/delay the game. You have to earn your outs. If there is 1 second left on the clock, the game can very well be over in most sports. In baseball, you could be down to your last strike, your last out and keep the game going.
"It ain't over until it's over". - Yogi Berra