Since the Fallout franchise is set in the United States of America, it makes sense that our national pastime would somehow be represented. Of course, this is not the USA of our timeline but it is close enough and they certainly had baseball. As far as we can tell, the sport had the same rules as it does in our timeline, or at least close enough. It certainly had the same equipment as you can find and pick up baseball bats, baseballs, bases, home plate, baseball caps, uniforms, baseball gloves, and umpire gear. Bats and balls are way more prevalent than the rest. Bats can be used as melee weapons just as they can be in our timeline.
Fallout 3
Baseball was a thing in Fallout 3 largely because it was set in Washington, DC (now known as The Capital Wasteland). Fallout 3 was in development as the current incarnation of the Washington Nationals was planned to make a permanent home in DC. In the world of Fallout, the team was instead known as the Capital Congressmen. Their stadium was not part of the map in the game so it could not be visited by the player. The Congressmen were known to regularly play (and beat) unnamed teams from Baltimore and Pennsylvania.
The only baseball fields in the game are the kind used for pickup games. The marked location is the Fordham Flash Memorial Field which looks like it was used for Little League games. The name refers to an actual baseball player/manager. His government name was Frankie Frisch and he played for the New York Giants (1919 - 1926) and the St. Louis Cardinals (1927 - 1937). He also managed several teams. He was elected to the Baseball Hall of Fame in 1947.
Fallout 4
Baseball was also alive and well in the Boston area where Fallout 4 takes place (although it is now called The Commonwealth). The clearest sign of that is that Fenway Park still stands in Boston. However, it was not the home of the Red Sox. Instead, the team was called the Swatting Sultans. This is obviously a reference to Babe Ruth who, among other nicknames, was known as The Sultan of Swat. It is mentioned in dialogue that the last World Series that ever happened was between the Swatting Sultans and an unnamed Texas team.
Fenway Park is no longer a baseball stadium and is instead repurposed as the largest settlement in the Commonwealth known as Diamond City. The park holds a lot of homes and a bustling market. The walls of the stadium and secured entrances and exits provide for a very safe location to live. The mayor’s office is in a skybox far above the market. The player can purchase the rights to a home named The Home Plate. There is also a bar built into the Boston dugout. The city’s security officers wear umpire gear.
The tradition of baseball is kept alive (kind of) by a shopkeeper named Moe Cronin. His name comes from two real-life baseball players. Joe Cronin was a famous baseball player for the Boston Red Sox (1935 - 1945), whose number (#4) was retired and can be seen hanging from the upper decks of Diamond City (and the real Fenway Park). Moe Berg was another Red Sox player (1935 - 1939) who was also a spy and a law school graduate. Moe Cronin sells baseball bats as weapons he calls “swatters”.
Poor Moe reveals how little the world still knows about baseball while sending the player on a mission. Cronin sends the player to the Westing Estate, the now-flooded home of a former Boston coach. The player must retrieve a signed baseball, a signed catcher’s mitt, and a signed baseball card. As you return them, Moe informs the player that two teams would meet on the field and try to beat each other to death with bats. The balls were given to the children of the deceased, baseball gloves were used to catch bullets, and baseball cards kept the kill stats of the players.
The player character in Fallout 4 happens to be somebody who was cryogenically frozen 200 years prior (in 2077) and remembers how baseball is actually played. If you decide to try to set Moe straight, he disagrees and thinks his rules are better. A lot of your companions agree with Moe.
There is also a rare baseball bat you can obtain by searching below the Jamaica Plains neighborhood called the 2076 World Series Baseball Bat in honor of the last World Series to be played. When swung, it has a small chance to launch the target like a home run.
Fallout: New Vegas and Fallout ‘76
I do not recall and I cannot find any evidence of any major baseball references in Fallout: New Vegas. Since the game takes place in what was once Nevada, it makes sense that baseball is not as present. Just as in our world, there must never have been a professional team in that state.
Similarly, Fallout ‘76 takes place where a professional baseball team has never played, West Virginia. However, it does reference two minor league teams playing out of Charleston and Monongah. I have not played the game yet but I do not see anywhere that references which team or teams that these would be farm teams for.