Aristotle, teaching in the Hellenistic period, did have an acknowledged cannon to consider. The three playwrights of fifth century Athens (Aeschylus, Sophocles and Euripides) had achieved a status through their productions at the Dionysia and subsequent revivals. Aristotle explicitly identifies Sophocles' Oedipus Rex as a paradigm, and it is from examples of the unities are frequently taken from this text.
The Unity of Action
"The plot, then, is the first principle, and, as it were, the soul of
a tragedy." (Poetics VI).
Having established the central importance of the plot, Aristotle goes on to explain how it operates in Attic tragedy. The Unity of Action insists that there is only a single plot, and no subplots. Oedipus follows the journey of one man towards self-revelation - and the social impact of his ignorance and discovery. The Oresteia plays out the development of human justice through Orestes revenge. The Suppliant Women is concerned with the resolution of an appeal for asylum.
The dynamism of Attic tragedy frequently depends on this single-minded focus on one plot. The limitations of the Athenian tragic format (three actors and a chorus, multiple productions in the same space on the same day) encouraged a purity of intention: overloading the actors with characters, exploring multiple ideas or non-choral interludes would stretch the capacity of the company.
The Unity of Time
"Tragedy endeavours, as far as possible, to confine itself to a single revolution of the sun, or but slightly to exceed this limit, whereas the Epic action has no limits of time. This, then, is a second point of difference; though at first the same freedom was admitted in Tragedy as in Epic poetry." (ibid, Book V)
At this point, Aristotle appears to limit the action to a twenty-four hour period. However, this is part of a contrast with the epic mode, exemplified by Homer's Iliad or Odyssey, and Aristotle recognises that this is not a universal rule. A hint that early tragedies were more discursive, and the caveat 'as far as possible' reveals that this is far from absolute.
Additionally, this is not written as a rule, but as an observation. Examining Oedipus, the passage of time is not as simple as suggested. There are choral interludes which cover passages of time - various characters are summoned, Creon goes to Delphi from Thebes (a journey of around fifteen hours on foot, according to Google Maps) and returns - and the appearance of a unity of time is actually carefully managed.
Indeed, tragedies were made that attempted a wider scope: he dismisses various Theseids and Hercleids because they tried to tell a hero's life story. Sadly, these have been lost, and their quality cannot be assessed, but tragedy clearly existed in the Hellenistic period which broke this unity.
The Unity of Place
That all of the action must take place in a single location is another quality perhaps dictated by the format of the Athenian theatre. Scene changes were not easily managed, and the shortness of each tragedy encourages a single set. However, the classical play texts do support this observation.