What was one of the greatest achievements of the eastern roman empire?

1. Construction of the Hagia Sofia is a big one. It still stands and is still very impressive.

2. Justinian's legal code. At the time he became Emperor, the laws of Rome had been through a few centuries of bloat and lost all legal precedents due to so many conflicting verdicts. He had it examined, updated and rewritten into a few texts. This paid the ground work and formed the basis for modern Western law.

3. Diplomacy, the Eastern Romans did not have the large reserves of manpower that the united Empire enjoyed and encountered many hostile neighbors over its life time. They used their wealth and influence to great effect to make allies with former enemies, to support claimants to create internal strife in a neighboring country, or get a third party to attack an enemy attacking them. Their Bureau of Barbarians was possibly the first of its kind.

4. Numerous Military Manuals. The Art of War has its place as the most famous, but the Eastern Romans wrote a lot too. Over the years they covered everything from war gear, logistics, formations, drills and training to common tactics and strategies against their neighbors.

5. Preservation of the histories, stories and arts of antiquity. This is a big one and the exodus of elites and scholars after the fall of Constantinople to Italy is often credited as one of the catalysts of the renaissance.