The problem would really lie in defining what is meant by "state." I say this because it is that kind of structural problem which ultimately arises; when you get big enough, it becomes impossible for a central government to rule an empire.
Different empires through history have approached this problem in different ways, broadly-speaking. Let's look at a few:
- The Romans
- The Mongols
- The British
- The Americans
The Romans enjoyed one of the oldest massive empires we know of; at its height, far before widespread literacy, mass media, or nationalist sentiment, the Roman Empire sprawled across three continents and commanded nearly half of the world's total human population. Worth mulling over a bit, I think. Anyway, this empire ruled from a central site (Rome), but to keep command of its far-flung peripheries, it delegated leadership to "client states" - that is, nominally in Egypt, for example, Egyptians might still rule at many levels of government, but there would be a Roman citizen near the top whose job it was to ensure that the wishes of the Empire were carried out by the local rulers. Later in its history, under immense pressure to deal with problems all along its very long borders, Rome split into two separate empires - the West and the East. This was, realistically, never going to be made whole again, because people are as people are, and no one would easily give up command of half the greatest empire the world had ever seen.
The Mongols came later and did things differently. They ended up controlling a much larger area of the earth, but not necessarily a larger part of the population; while a larger number of people possibly lived on the lands they claimed, the Mongols didn't really ask a whole lot of those populations once they owned them - they killed quite a few and did relatively little with them once they had them, save perhaps levy taxes, to be carried out by, again, local leadership, this time without even necessarily a Mongol citizen in place - instead the threat of the Mongol Horde was deemed sufficient to coerce behavior friendly to the empire. The Mongols ruled via these mobile armies first and foremost, and it makes clear that you can control vast territories and resources without "you" (that is, the citizens of your empire proper) being very large. The technology of the horse makes this possible for those willing to exploit it fully.
The British were exploiting another conveyance - the sailing ship - to rule their far-flung empire of colonial holdings - and again, the question is whether you want to consider all their lands and peoples properly part of the "British Empire" - because they might have "ruled" them, but under the auspices of things like the East India Company, a corporate affair that pretty much managed the Indian subcontinent for them for much of its history within the British Empire.
The British are fascinating because I think their empire in many ways models the problems a far-flung spacefaring empire would have to overcome - some of the greatest problems they experienced were the expense of projecting their power overseas, the loyalty of their colonies to their empire, and the communications troubles of such distant holdings. This is a polity that took an entire continent and couldn't do anything with it other than basically use it as a prison for a while. Again, whether we consider all Australia properly "owned and controlled" by the British at all points they claim to have owned it is debatable in my mind.
The United States of America, perhaps in response to the the problems of their British parents, had this idea that smaller confederated states should coexist under a federal, central government, which would effectively facilitate the cooperation of the state governments, which were themselves the thing seen more close to countries in their day. The "nation" of states which you sometimes hear the founding fathers of American government refer to encompasses this plurality, and indeed you can look at the diplomatic, military, and economic dealings of other countries and empires in the world with the early USA and see that there was considerable confusion about whether they were a group of independent states, or whether they were provinces of a larger federal government. That question eventually got resolved in a pretty nasty fight about a different but related problem, showing that even this method of controlling vast numbers of people and tracts of land has its pitfalls - namely, that if you enfranchise most of your highly heterogeneous population, there will be significant and empire-ending disagreements unless the confederation of states idea is abandoned in favor of a much more powerful central government. This then suffers the same problems as large, centralized empires of the past, but luckily America arises in a period of vastly improved conveyances (steam engines and telegraphs), and these technologies, along with high literacy and "universal" suffrage, allow it to mold a sense of national belonging throughout its population - enforcing its empire from the bottom up more than the top down. Indeed, this worked so well that it became the ultimate model of the modern nation-state; Europe especially quickly succumbed to this nationalism soon thereafter; nothing stands in the way of an empire whose entire citizenry is employed in the cause. This is an excellent reason to reconsider how big an empire truly is based on its method of control; though Mongolia might field a mighty army in terms of raw power and size compared to other feudal states, nothing compares to the kind of massive, million-strong citizen army France could field with its nationalist instincts in full swing, despite a much lower total population.
Ultimately, limits on coherent size will depend on the nature of the state in question. A "Mongolia" rule-through-fear and not-caring-ness might yield a very large empire in terms of size and resources, but Nationalist states with modern communications technologies would be the more powerful, more truly populous alternative, however much land they actually owned.
Finally, a note on technologies - at a certain point, you will run up against the problem of latency. Even with the internet, you are fighting, ultimately, the speed of light, and it becomes difficult to impossible to communicate with colonies separated from the federal control center without years-long lapses in judgment. The natural response to this will be to delegate, as the Romans did when they were dealing with the latencies inherent to foot and horse travel, and then from those small fractures will arise great schisms which will shake your nation-states apart one after the other, and the surface of your world will become an infinite Balkans, united only by the effective reach of light speed communications and the rate of change inherent to the forces of disunity they must struggle against to remain whole.
Empires might consist of many billions of people and many dozen Russias' worth of landmass, but it is difficult to imagine an order of magnitude larger than that working for very long.