In classical sources, the notion of state does not exist in the meaning it has now. As you probably know, a state has several definitions :

• A sociological one : a State is a group of person living on a given territory with a given government.
• An organizational one : a State is the form of organization retained by a society, a collective that gives itself a set of rules organizing itself.
• A legal one^1 : a State is a set of powers of authority and constraint, exerted by the Nation upon its citizens so that the common good (intérêt général, of which law and order, the idea of ordre public is part) can prevail. In the specific field of international law, States are the most frequently encountered legal person.

The two first definitions arose at least as early as the late Renaissance : in the Dictionnaire de l'Académie Française's first edition (1696), both definitions (a State is a group of person on a given territory, with some idea of a shared identity, and a State as their mode of organization) are present. If we look further at where this definition comes from, we can find that Machiavelli is probably one of the very first to use stato as meaning "the political unity of a given people, beyond the eventual changes of governments or forms of government" (see for instance Il Principe^2). As you can see, Machiavel's definition does not require a notion of defined geographical boundaries (fortunately, as state boundaries were quite shifting in the Italy of his times).

Before Machiavel, the word state does not seem to find a translation maintaining a close enough definition. To quote Thomas Fleiner-Gerster in his Théorie Général de l'Etat, "L’Etat moderne en Europe et aux Etats-Unis est né d’une évolution séculaire à partir de l’ordre social fortement hiérarchisé du Moyen Age." (the contemporary notion of State in Europe and the US is born of a slow evolution over centuries, from the highly hierarchised social order of the Middle Ages).

This does not mean that a people like the Romans would not have had some idea that there was a Roman state (in the meaning of a set of institutions put in place so that the common good, and in particular law and order, would prevail) : the word for this was res publica (the public thing, literally). They also had some idea that they were a specific polity, inscribed in a territory, with specific institutions and a shared identity (and a shared destiny, could we say). In that sense, E. Renan^3 considered them one of the very first nations in the modern sense ; the Romans nonetheless used the name of a Nation for any organized group with a shared language and/or identity : in the Commentaries on the War in Gaul, J. Caesar describes the Gauls as divided in several nations, as well as the Germans ("The Germans, at last compelled to fight, went out of their camps and placed themselves in the order of their nations, Harudii, Marcomani, Triboqui, Vangiones, Nemeti, Sedusianii, Suevii", op cit, I.51). Those nations were not specifically attached to a territory, and could be considered a legal person for purposes of treaties : for instance see the various treaties with the Haedui in Gaul).

Parallelly, the Romans has the idea of general geographic areas encompassed by broadly natural borders (for instance Gaul, or Germany, or Africa), in which could be several nations.

As a conclusion, for the Romans, there was no ancient conception of a state. But to them, the peoples they dealt with were nationes, without anything more than loose territorial attachments (for example, the migrations of the Helvetii at the beginning of the war of the Gauls didn't lead them to stop being an Helvete nation).

^1 : Nota : this definition is heavily tainted by concepts derived from French constitutional law, and generally from the Romano-germanic legal tradition : in the English (and US) legal tradition, a different definition would probably be more accurate.

^2 : Thomas Fleiner-Gerster, Théorie Général de l'Etat

^3 : E. Renan, Qu'est-ce qu'une nation ?, 1882.