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I know that there is probably not just one answer, so let's restrict the question to the golden era of 4th/5th century BC, in Athens.
If they wanted to refer to Homer's time, who lived 300 years before them, how would they have done it?
Your question could mean a couple of different things.
The ancient Greeks didn't date historical events by use of a calendar. They had local calendars to regulate events during the year, but:
By contrast [to the modern calendar], the Attic calendar had little interest in ordering the sequence of years. As in other Greek cities, the name of one of the yearly magistrates, at Athens known as the eponymous archon, was used to identify the year in relation to others.
-Wikipedia, Attic Calendar
This is analogous to the system used by other ancient peoples, such as the Egyptians, who would date an event according to the regnal year of the current pharaoh.
But that refers to individual years, not "eras".
When you ask about the "era of Homer", you seem to be referring to the time Homer was active, and not the time of the things he was writing about, so we'll address that next. The time when Homer "lived" is a fraught thing to talk about, because it's not clear such a person as "Homer" ever lived. Even the ancient Greeks disagreed on when that was, and even then some questioned if Homer existed.
One of the few surviving estimates of that time is in Herodotus, in one of the rare places he admits one of his inventions is his opinion:
Whence the gods severally sprang, whether or no they had all existed from eternity, what forms they bore- these are questions of which the Greeks knew nothing until the other day, so to speak. For Homer and Hesiod were the first to compose Theogonies, and give the gods their epithets, to allot them their several offices and occupations, and describe their forms; and they lived but four hundred years before my time, as I believe. As for the poets who are thought by some to be earlier than these, they are, in my judgment, decidedly later writers. In these matters I have the authority of the priestesses of Dodona for the former portion of my statements; what I have said of Homer and Hesiod is my own opinion.
-Herodotus, Histories, book 2 pp. 53
1911 Brittanica concludes the following from Herodotus's wording:
Of the date of Homer probably no record, real or pretended, ever existed. Herodotus (ii. 53) maintains that Hesiod and Homer lived not more than 400 years before his own time, consequently not much before 850 B.C. From the controversial tone in which he expresses himself it is evident that others had made Homer more ancient; and accordingly the dates given by later authorities, though very various, generally fall within the 10th and 11th centuries B.C. But none of these statements has any claim to the character of external evidence.
Notice that Herodotus doesn't give a special name to this period, only a number of years in the past. And it's a vague, extremely round estimate at that. Nevertheless, some add 400 to the estimated time Herodotus wrote his Histories (around 450 BCE), to get 850 BCE. This is during the tenure of the Athenian archon Pherecles (864-845 BCE), altough we have no way to conclude Homer lived during this period.
Both estimates place Homer in the period we call the Greek Dark Age. Hesiod told the story of five "Ages of Man" in his Works and Days, although ancient Greek historians didn't really use them as a framework for historiography.
The 10th or 11th centuries BCE estimate places Homer close to the Heroic Age of the Iliad and Odyssey.
But when earth had covered this [Bronze] generation also, Zeus the son of Cronos made yet another, the fourth, upon the fruitful earth, which was nobler and more righteous, a god-like race of hero-men who are called demi-gods, the race before our own, throughout the boundless earth. Grim war and dread battle destroyed a part of them, some in the land of Cadmus at seven-gated Thebes when they fought for the flocks of Oedipus, and some, when it had brought them in ships over the great sea gulf to Troy for rich-haired Helen's sake: there death's end enshrouded a part of them.
Hesiod, Works and Days, 155-166
However, if Herodotus is right, then Homer is in the same time as Hesiod, the Age of Iron:
Thereafter, would that I were not among the men of the fifth generation, but either had died before or been born afterwards. For now truly is a race of iron, and men never rest from labor and sorrow by day, and from perishing by night; and the gods shall lay sore trouble upon them.
-Hesiod, Works and Days, lines 174-175
If, instead of the time Homer lived, you actually mean the time of the things Homer wrote about, then this would have been during the Heroic Age.