Eastern Rome (wrongly called Byzantine, something most people then wouldn't even have known what Byzantium is) went on and was fully the Rome Empire. It never actually fell. All the learning and information was continued for the most part.

Sadly even they had a lot of problems in the 'dark age' (not a good name either). First their was the Justinian Plague and then they had a lot of civil war and conflict over Iconoclasm.

In general because of how Europe developed, and how the Renaissance happened in terms of intellectual history a lot of this history is only remember by very few people.

By 1000AD the Roman Empire was still going strong, being the most powerful state in along the Mediterranean.

Edit:

Check 'The History of Byzantium' Podcast, a follow on from the 'History of Rome' Podcast but by a different person.

Thersites the Historian has lot of good content on yt, for example https://www.youtube.com/playlist?list=PLEQru6POYgetKOdzDYaLy...