You're living in Constantinople in the 10th century. Do you regard tales of Rome's expansion over 1000 years ago as a factual account of prior events?

Yes. Citizens of Constantinople were conscious of their long history. It's hard to get a "man on the street" perspective, but there is a source compiled in the 10th century called the Patria of Constantinople, a sort of secular guide to the city and its monuments meant for popular reading. It heavily borrows from much older sources, but shows that 10th century Byzantines were conversant with the past. It begins with a passage that rather succinctly answers your question (From the 5th century):

When 362 years had passed since the sole reign of the Caesar Augustus in the elder Rome, and her fortunes were already coming to an end, Constantine the son of Constantius took over the scepters and established the new Rome, ordering that it should be equal in rank to the first. For after having been ruled by tyrants and kings, and often having been governed by the ways of aristocracy and democracy, it finally achieved is present greatness. So we should tell how it originated in the beginning and by whom it was settled, using the old poets and writers as our basis.

Or more like an apocryphal tale of an empire that spanned the known world before falling into terminal decline?

Yes and no. Ideologically it was important to emphasize that Rome had fallen into disrepute in some way, especially once it lay outside of the empire's borders. Although it became increasingly untenable, Byzantine Emperors claimed the same sort of universal authority that the emperors of antiquity did. So Rome the city fell, not Rome the Empire. An Italian Lombard ambassador and partisan of the Holy Roman Emperor, Liudprand of Cremona, traveled to Constantinople in the late 10th century and was taken to task for a letter sent by the Pope addressed to the "Emperor of the Greeks", which infuriated the emperor. Luidprand quotes a Byzantine official's response which, although from a fierce partisan like himself, more or less illustrates the Byzantine idea of their heritage:

"Then listen: the empty-headed and bungling pope is ignorant of the fact that holy Constantine translated the imperial symbols here, and brought the entire senate and the whole Roman knighthood, and left at Rome only the lowly dependents, that is, fishers, food-peddlers, bird-hunters, bastards, plebeians, and slaves."