Is there any specific date and time when William Shakespeare's play, A Midsummer Night's Dream, was set?

This is a difficult question to answer authoritatively,
because I have seen numerous productions of this excellent play with settings ranging
from some kind of Athenian golden age in medieval times to modern-day settings with all
of the characters in modern dress. We always need to remember when studying Shakespeare
that all he left us with is the text. Whilst there are a few pointers in terms of
location and time in that text, so much else is necessarily a choice and decision of the
director who is staging the production of that play. This play could equally be set in
modern day America as it could be set in feudal China, for example, and still be just as
relevant. Consider the film version of Romeo and Juliet that sets
it in modern-day LA rather than in medieval Italian Verona for an excellent example of
this.

Thus, when we look at the play, the fact that Theseus
is called "Duke of Athens" does point towards some kind of medieval setting in Greece,
but it is vital not to feel bound or limited by that. If you are trying to come up with
a setting to this play, you can go wild as far as your imagination will take you, as
long as your setting can roughly be linked in to the themes of the
play.