The unities of time, place, and action were derived from Aristotle's Poetics by an Italian theorist, Lodovico Castelvetro, in 1570.

In Poetics, Aristotle proposes the unities of time and action (unity of place isn't mentioned in Poetics) as guidelines rather than actual rules, which are based on what Aristotle observed in Greek tragic plays—not what he intended to impose on the writing of tragic plays. Castelvetro and French classical playwrights like Pierre Corneille, Jean Racine and Jean-Baptiste Poquelin (better known as Molière) interpreted these observations as requirements and adhered to these unities in their plays.

According to Castelvetro, the unity of time imposed a twenty-four hour time limit on the action of the play, the unity of place meant that the action of the play should occur in a single location, and the action of the play be restricted to a single, unified plot line:

Tragedy endeavors, as far as possible, to confine itself to a single revolution of the sun, or but slightly to exceed this limit. (Poetics, part 5)

In Oedipus Rex, the action of the play takes place in a single day. Even events which would take longer than a day, such as Creon's trip to the Oracle of Apollo at Delphi, are reported rather than actually enacted in the play.

Aristotle makes no mention whatsoever about a unity of place in Poetics. This unity was apparently devised by Cornielle as an outgrowth of the other two unities and Castelvetro's writings. The unity of place can also be logically assumed from the actual construction of an ancient Greek theatre, which limited the action of the play to what could reasonably occur in a single location. Nevertheless, Oedipus Rex conforms to this unity, in that the entire action of the play occurs in front of the palace at Thebes.

The only unity which Aristotle seems to have insisted on was the unity of plot, which he considered an integral part of a tragic play:

The Plot, then, is the first principle, and, as it were, the soul of a tragedy . . . [T]he plot, being an imitation of an action, must imitate one action and that a whole, the structural union of the parts being such that, if any one of them is displaced or removed, the whole will be disjointed and disturbed. (Poetics, part 6–8)

Aristotle further imposed on the unity of plot that the action, the events of the play, must have a cause-and-effect relationship:

Tragedy is an imitation not only of a complete action, but of events inspiring fear or pity. Such an effect is best produced when the events come on us by surprise; and the effect is heightened when, at the same time, they follow as cause and effect. . . . Plots, therefore, constructed on these principles are necessarily the best. (Poetics, part 9)

Oedipus Rex is constructed around and driven forward by a single objective: find the person who killed Laius. Every event in the play is focused on this objective, and nothing that occurs in the play is extraneous to this objective. The sometimes surprising events of the play follow a strict cause-and-effect relationship that lead inexorably to the conclusion that the person who killed Laius is Oedipus himself.

As a previous educator has rightly noted, all three of Aristotle's dramatic unities are present in Oedipus Rex. Indeed, for Aristotle, Sophocles's masterpiece was the paradigm example of how a play should be constructed. The entire action of the play takes place over the course of twenty-four hours in the royal palace at Thebes. The action is itself a unity as it is focused entirely on one single theme—Oedipus's investigation of Laius's death and the terrible truth it reveals.

The three unities, when combined, allow complete economy of expression; everything is extraordinarily concentrated and precise. There are no subplots, no comic relief; all our attention is focused on Oedipus and his terrible fate. At the same time, the simple presence of the three unities is not in itself enough to constitute a great play. Aristotle's unities should be seen rather as a structure, a foundation on which the playwright builds the various elements of his or her drama.

Aristotle's three unities are unity of place, unity of time, and unity of subject.  Oedipus Rex, Sophocles's tragedy, achieves all three of the Aristotelian unities.  

First, it achieves unity of place because all of the action happens only in one place: in front of Oedipus and Jocasta's royal residence.  Whenever the characters need information, they call someone to them rather than leaving this place; or, when events occur inside, someone comes out to tell the characters.

Second, the play adheres to the unity of time because everything takes place within twenty four hours.  For example, instead of having Oedipus decide to send Creon to the oracle during the course of the play, and then having to wait several days for him to return with his news, Sophocles has Oedipus think ahead and send Creon several days before anyone suggests it; that way, Creon can return at just the right time to deliver his information so that the play can promptly continue.

Third, the play achieves unity of subject by only focusing on Oedipus's tragedy.  There are no subplots or story lines that focus on other characters to distract us from Oedipus's immense pride and tragic end.