The Epic of Gilgamesh was written ca. 2000 BCE about a Babylonian king who lived ca. 2700 BCE. The most basic evidence for its antiquity is the fact that the story was discovered during an archaeological dig at the ancient city of Uruk in what is now Iraq. The story was "written" on clay tablets in cuneiform. The story is sold old that Gilgamesh as a character predates the Greek gods.
One aspect of the epic that sets it in ancient Babylon is the people's polytheism; in fact, Gilgamesh himself is said to be partly divine.
Gilgamesh's divine nature adds to another element of ancient stories: A superhuman who is so powerful that the gods must make a creature to try to destroy him. In this story, the gods create and send Enkidu against Gilgamesh.
Another element that sets the epic in ancient times is the story of the flood. Many biblical scholars today believe that the Old Testament story of Noah's flood is actually based on the Epic of Gilgamesh. The Israelites had been conquered and taken into captivity by the Babylonians, and so they would have come to know the Babylonians myths and later generations would adapt them as their own (I'm not sure it's not the other way around!).
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An interesting question. I'd have to say "briefly," and "as the universal human condition." By that I mean, there are references in the poem to things we now see as period specific, such as the temple prostitutes or the layout of the city, but there's not much attention paid to it. This isn't primarily a poem about the current society. Instead, a few things are sketched into being; it is assumed that the kingship comes from a divine lineage, there is a pantheon of gods, there is trade and travel, and so on.
While it is true that sharing cultural information is not the purpose of the story, still there is much that can be gleaned from its pages (or should I say tablets) about Sumerian culture. We learn about their polytheistic religion and about there view of life after death. We learn about their political organizations: the city-states and the kings. We learn that Gilgamesh returns home and WRITES about his adventure. That's huge. We also learn about the role of various societal elements- women were supportive, not very powerful, priests and priestesses were respected, etc.