To what extent did the Byzantine Empire embody the Roman Empire after the Western Roman Empire fell? Say, in the year 1000 CE, did the Byzantines have running aqueducts and other staples of Roman society, or were they basically just another Medieval society by that point?
They had a much more powerful central state apparatus and a much less powerful landed aristocracy than any other polity of western Eurasia during the Middle Ages outside of the Middle East. As of 1000 AD, they were still (after over six hundred years) on the gold standard. Constantinople (population ~100K) was still the largest city in the Mediterranean other than Alexandria (population ~300K). As a result, the Aqueduct of Valens was still very much repaired and running c. 1000 AD. The Byzantine Empire also preserved far, far more ancient texts than did the Medieval polities of Western Europe or the Muslim world, despite experiencing a catastrophic urban recession after the cutoff from Egypt c. 640. As a result of these factors, Kaldellis, one of the leading scholars of the period, argues "Byzantium was not Medieval"; that is, it did not have the peculiar characteristics of medieval northwestern European polities. However, there were huge cultural transformations during the Late Roman Empire (say 235 to 600), which should not be ignored. Of all the ubiquitous and notable classical imperial structures, only baths were still being built by the mid-sixth century, for instance, and aristocratic wealth was largely moved away from public display.