There is no certain date. Most scholars place Homer in the eighth century BC, around 750 BC, and the Odyssey perhaps a little after the Iliad. Herodotus, in the fifth century BC, said that Homer had lived some four hundred years before him. The war the poems narrate would be much earlier: the destroyed Troy identified by the excavations is dated to around 1200 BC.
The Homeric period is the name historians give to the centuries from the fall of the Mycenaean palaces, around 1200 to 1100 BC, to the eighth century BC. It is also called the Greek Dark Ages, because the writing of the palaces, Linear B, was lost, the cities shrank and trade declined. Society was organized around families and local chiefs, and much of what we imagine about it comes from the poems themselves, hence the name.
At the end of this period, in the eighth century BC, Greece rises again: the Greeks adapt the alphabet of the Phoenicians, the city-states are born, the colonization of the Mediterranean begins and the first Olympic Games are counted, in 776 BC. It is in this world that the poems take the form we know.
That is why the Iliad and the Odyssey mix periods. They keep memories of the Mycenaean world, such as Mycenae rich in gold and the boar's-tusk helmet, and cover them with customs of the poet's time, centuries later.
On Clavis: The Iliad, The Odyssey.