One thing we know is that Shakespeare is presenting in the play a very ancient civilization. We know this because the ruler over Athens is Duke Theseus, which can also refer to the mythical King Theseus who founded Athens and married Hippolytus, as portrayed through the character Hippolyta. Since Athens started as a monarchy and much later became a democracy around 508 BC, we know that the play is set in Ancient Greece. We also know that their political system is that of a monarchy.
We also learn a few things about the characters' culture. We learn from the opening scene that in their culture men are considered superior while women must be subordinate. In the opening scene, Egeus brings his daughter Hermia before the duke because she refuses to marry Demetrius, whom he has commanded her to marry, showing us that their society considers it to be a crime for a woman to disobey her father's will. We especially see this when, after hearing Egeus's plea, Theseus turns to Hermia and reminds her that she is expected to treat her father with reverence, like a god, as we see in his line, "To you your father should be as a god" (I.i.49). Another thing we know about the Athenian culture presented in this play is that Athenians place great value on social rank and wealth. We learn this when we see Lysander protest against Egeus's decision to want Hermia to marry Demetrius instead of himself considering that Lysander is equal to Demetrius in social rank and even wealthier than Demetrius, as we see in Lysander's lines:
I am, my lord, as well derived as he,
As well possess'd; my love is more than his;
My fortunes every way as fairly rank'd,
if not with vantage, as Demetrius. (101-104).
The word "vantage" in this passage can be translated as "even more"; hence, Lysander is arguing that he is just as wealthy as Demetrius if not more so. This conversation of social rank and wealth shows us just how important both attributes were in this ancient Athenian society.
We also learn something about Athenian law from this scene. We learn that they have an ancient law granting permission for a father to kill his daughter if she disobeys him or to send her to a convent, as we see when Egeus petitions Theseus for permission to inflict "the ancient privilege of Athens" and to "dispose of" Hermia in the way Egeus sees fit, either through marriage or through death (42-46).