The very concept of "Age of" is flawed. If one hears phrases like stone age or dark ages, which are almost about the only two ages an historian would use seriously, you have instantly negative connotations about the timeframe to discuss.
That is not to say that "ages" are not a well established, popular and even classic concept:
The Ages of Man are the stages of human existence on the Earth according to Greek mythology and its subsequent Roman interpretation.
Very much like in Hesiod's ages of man:
- Golden Age – The Golden Age is the only age that falls within the rule of Cronus. Created by the immortals who live on Olympus, these humans were said to live among the gods, and freely mingled with them. Peace and harmony prevailed during this age. Humans did not have to work to feed themselves, for the earth provided food in abundance. They lived to a very old age but with a youthful appearance and eventually died peacefully. Their spirits live on as "guardians". Plato in Cratylus (397e) recounts the golden race of men who came first. He clarifies that Hesiod did not mean men literally made of gold, but good and noble. He describes these men as daemons upon the earth. Since δαίμονες (daimones) is derived from δαήμονες (daēmones, meaning knowing or wise), they are beneficent, preventing ills, and guardians of mortals.
- Silver Age – The Silver Age and every age that follows fall within the rule of Cronus's successor and son, Zeus. Men in the Silver age lived for one hundred years under the dominion of their mothers. They lived only a short time as grown adults, and spent that time in strife with one another. During this Age men refused to worship the gods and Zeus destroyed them for their impiety. After death, humans of this age became "blessed spirits" of the underworld.
- Bronze Age – Men of the Bronze Age were hardened and tough, as war was their purpose and passion. Zeus created these humans out of the ash tree. Their armor was forged of bronze, as were their homes, and tools. The men of this Age were undone by their own violent ways and left no named spirits; instead, they dwell in the "dank house of Hades". This Age came to an end with the flood of Deucalion.
- Heroic Age – The Heroic Age is the one age that does not correspond with any metal. It is also the only age that improves upon the age it follows. It was the heroes of this Age who fought at Thebes and Troy. This race of humans died and went to Elysium.
- Iron Age – Hesiod finds himself in the Iron Age. During this age humans live an existence of toil and misery. Children dishonor their parents, brother fights with brother and the social contract between guest and host (xenia) is forgotten. During this age might makes right, and bad men use lies to be thought good. At the height of this age, humans no longer feel shame or indignation at wrongdoing; babies will be born with gray hair and the gods will have completely forsaken humanity: "there will be no help against evil."
But that does not make it useful for historical discussion. Not because of mythological content. But because it is Hesiod's attempt to construct a useable past, or as we would call it: writing history. In our contemporary understanding we cannot use the same terms to describe the same periods of time. But the theme of nostalgic past and decay for the present and future very well resonates today. But that makes it more about a certain world view, philosophy or just politics. From Hesiod's description, we for sure still live in a heroic age, and are discontent with it.
In contemporary – owing to the nature of the question I hesitate t call it modern – historiography, we need a different approach, if we want to keep is on a level of scientific investigation.
That is called periodization:
Periodization is the process or study of categorizing the past into discrete, quantified named blocks of time2 in order to facilitate the study and analysis of history. This results in descriptive abstractions that provide convenient terms for periods of time with relatively stable characteristics. However, determining the precise beginning and ending to any "period" is often arbitrary. It has changed over time in history.
To the extent that history is continuous and ungeneralizable, all systems of periodization are more or less arbitrary. Yet without named periods, however clumsy or imprecise, past time would be nothing more than scattered events without a framework to help us understand them. Nations, cultures, families, and even individuals, each with their different remembered histories, are constantly engaged in imposing overlapping, often unsystematized, schemes of temporal periodization; periodizing labels are continually challenged and redefined, but once established, a period "brand" is so convenient that many are very hard to shake off.
That gives the problem of arbitrariness: Does it make sense to speak of medieval China? Does it make sense to apply concepts fitting very well into Western/European history to Africa? Sometimes it does. Most times it doesn't.
To really come to terms on a general level with the question, we might need to grab some inspiration from art history. There, certain styles are useful for periodization. This is proto-geometric pottery, this is gothic architecture, and this is impressionist painting style. When were these styles named as such? Seldom, when they were "in". At the earliest when they were on the way out. Therefore: it is only then a style if you do not have it any more.
See for example this currently unresolved HistorySE question: What did they call Gothic and Baroque architecture before the modern terms came into use?. The 'style' originated in the 'high middle ages' in France and we now quite clearly recognise churches and other buildings from that time as gothic. They apparently didn't see it that way:
The Gothic style began to be described as outdated, ugly and even barbaric. The term "Gothic" was first used as a pejorative description. Giorgio Vasari used the term "barbarous German style" in his 1550 Lives of the Artists to describe what is now considered the Gothic style.
WP: Gothic architecture
In other words: when a period, or age if you will, starts, ends or what it defines, cannot be seen by us.
The "hellenistic period":
covers the period of Mediterranean history between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire as signified by the Battle of Actium in 31 BC
The word originated from the German term hellenistisch, from Ancient Greek Ἑλληνιστής (Hellēnistḗs, "one who uses the Greek language"), from Ἑλλάς (Hellás, "Greece"); as if "Hellenist" + "ic".
"Hellenistic" is a modern word and a 19th-century concept; the idea of a Hellenistic period did not exist in Ancient Greece.
This is the job of future historians. And they will argue about it with the most joyous and bitter disagreements.
We might still define 'an age' for us and to us, believing to be living in it. That this classification will survive even the next generation is highly unlikely.