The play is overtly set in ancient Greece, a time and place far distant from the Elizabethan theater in which the play was performed. The ancient Greece of the play is not that of classical antiquity but a mythological age; Theseus was a legendary hero of Athens, not a real person, and in classical Greek drama he is placed in a distant past.
The first scene, that of the preparations for the wedding of Theseus and Hippolyta, takes place in this legendary Athenian setting.
The second setting is the woods. The four young lovers flee to a wood outside Athens. This wood, however, actually contains Puck (or Robin Goodfellow), Oberon, and Titania, figures from French and British medieval folklore. The dream wood, in fact, has a British feel to it, despite theoretically being outside Athens. The characters all return to Athens for the weddings at the end.
The most important thing to note here is what Puck says at the end of the play:
That you have but slumbered here
While these visions did appear.
In other words, the play itself has the character of a dream, and the setting, like that of a dream, is not realistic but fantastic, blending ancient and medieval and Greek and British elements for atmosphere.