Approximately when is the story of "Boku no Hero Academia" set?
Unless cannon ever gives us an exact year, it’s impossible to say for sure. Thematically though, the story takes place 20 minutes into the future

Technology and culture is almost identical to today, except that there are people with superpowers and has been for generations. The problems that the character’s face, and society as a whole face, aren’t that different from the problems we face in real life (except dialed up to 11 because of the presence of Quirks, and the fact that this is an action work of fiction).