Has there ever been a father and son in the NFL/NBA/NHL/MLB who were both active players at the same time?
Absolutely. Ken Griffey Jr. and Ken Griffey Sr. played together in 1990 for the Seattle Mariners. They once hit back to back home runs in a game.
They played about a month together (8/31 to the end of the season).
In Hockey, Gordie Howe played with his sons Mark and Marty in 1979-80 for the Hartford Whalers. Wikipedia has a nice page with a list of familial relations in the NHL (they seem to have more than most leagues, probably because the sport is less popular so more likely for familial relations to be successful). The direct link to the Howes’ is at Legends of Hockey.
In the NFL, it doesn’t look like anyone really did in the modern era due to the very short career length compared to other sports (except for kickers); in the 1920s, Ted and Charles Nesser played together, Ted as player-coach, and I don’t doubt a few more similar occurrences would be found if you looked through the 10s and 20s.
George Wilson coached his son (also George) for the 1966 Miami Dolphins, the only example of that I found. Ed and Brad Budde were the closest to playing with each other from the list I saw, Ed playing 1963-1976 for the Chiefs and Brad joining the Chiefs in 1980-1986. This is from a list of father-son relationships from the Pro Football HOF.
From the comments, Leatherwing noted that Wikipedia also has a good NBA Father/Son page; Kobe Bryant nearly overlapped with Bean, his father (1992/1996).