I'm not sure if "weak" is the right word to describe them.

The Romans were a victim of their own success. They were so rich that everyone wanted a piece of the pie, especially internally.

Corruption was normal, and their emphasis on military conquest meant that at any given time you had several very powerful private citizens with armies loyal to them (and their money) trying to carve out their own empire. They were also willing to hire people outside the Roman Empire.

The Romans conquered other people, and there shouldn't be any surprise when those other people try to conquer them back.

So the Byzantines inherited all of this, and they were constantly in some civil war or some coup or they were attacked by multiple enemies at once. Of course, they also were competing with other empires (especially the Iranian-based ones) for the same resources and territories.