Time travel stories have been around since the days of the ancient Greeks - Plato's dialogue, the "Timeaus", explores the possibility of travel to other times, as well as the idea of immortality through time travel. Written around 360 BC, this story is likely the oldest time travel story in existence, and is still relevant today.
In the ancient Roman period, Lucius Apuleius wrote his famous novel, "The Golden Ass", which was an early version of the time travel story. This tale dates back to around the 2nd century, and includes the theme of accidentally travelling through time as a result of a magical potion.
In the early Middle Ages, the great philosopher and theologian Al-Ghazali wrote "The Incoherence of the Philosophers", another work that involved time travel elements. This work, from the 11th century, is a classic of Islamic philosophy and it approaches the concept of time travel from a spiritual viewpoint.
Finally, the work most people are probably familiar with is the classic "The Time Machine" by H.G. Wells, which dates to 1895. This story is still considered a classic of science fiction and is probably the most widely known time travel story in existence.
So, there is no definitive answer to the question of the oldest time travel story, but the evidence points to Plato's "Timeaus" as the likely candidate. This story, written in ancient Greece around 360 BC, is an early exploration of the possibilities of time travel and still resonates with readers today.