When did this story take place?

No one knows precisely when The Odyssey was actually written. However, most scholars agree that it was probably composed some time in the eighth century BC. The setting of the poem—although we cannot be absolutely sure—is most likely set during the thirteenth century BC. This is known among historians as the Bronze Age. This was an appropriate historical setting for The Odyssey, as the ancient Greeks regarded it warmly as a golden age, a time of great heroic deeds.
Although the events of the poem are fictitious, archaeological evidence appears to suggest that some kind of war between the Achaeans and the Trojans was fought around this time. Additionally, as the action of The Odyssey takes place not long after the Trojan War has ended, it is safe to assume that this would be the historical era in which the events unfold.
https://www.ancient.eu/Odyssey/