I know the concept of "Roman citizenship" features prominently in accounts of Roman history, especially when the relation of the Romans with subjugated people is described. This means that the Romans had an at least implicit idea of what constituted statehood. How was the Roman state formulated and how did one acquire participation therein? Did the Romans treat their statehood by the tripartite definition we use today (territory, people, effective power) or was it a cultural or ethnic conception? Did the Romans generalize their concept of a state to form a category that included other polities that we recognize as such today and that they would have known of? I.e. could a Roman have spoken intelligibly about Rome and the Persian Empire being both "states", in the same way as we know that Italy and Iran are both "states"?
The closest would probably be Romans referring to the res publica or "the public affair". Cicero talks about the republic extensively in De Re Publica. The following passage is from chapter 9:
Those apologies, therefore, in which men take refuge as an excuse for their devoting themselves with more plausibility to mere inactivity do certainly not deserve to be listened to; when, for instance, they tell us that those who meddle with public affairs are generally good-for-nothing men, with whom it is discreditable to be compared, and miserable and dangerous to contend, especially when the multitude is in an excited state. On which account it is not the part of a wise man to take the reins, since he cannot restrain the insane and unregulated movements of the common people. Nor is it becoming to a man of liberal birth, say they, thus to contend with such vile and unrefined antagonists, or to subject one’s self to the lashings of contumely, or to put one’s self in the way of injuries which ought not to be borne by a wise man. As if to a virtuous, brave, and magnanimous man there could be a juster reason for seeking the government than this—to avoid being subjected to worthless men, and to prevent the Commonwealth from being torn to pieces by them; when, even if they were then desirous to save her, they would not have the power.
The phrases "public affairs" and "Commonwealth" are both translated from "rem publicam" and it is clear that, at least in the second case, the commonwealth refers to something greater than just the government, which we can consider nowadays to be a reference to the state. Other authors, like Pliny, Tacitus, Cato, Suetonius, and Augustine, talk about "res publica" referring to various things, some of which we can interpret to being the state. So, I would argue that, it is possible that Romans had some contemporaneous conception of what a state was. Their conception centered around, naturally, those things that were in the public affairs, which includes the ideas of citizenship, the public institutions (like the Senate and Assemblies), and the constitution ("res publica" or variants can sometimes be translated as constitution as well).
However, it was not exactly our modern conception. Our modern conception probably stems first from the definition of Westphalian Sovereignty after the 30 Years' War, which developed in the following 300+ years to eventually come to our modern understanding.