How did the byzantine empire last so long?

There were many factors that allowed the Byzantine Empire to last a 1000 years after the end of the Roman Empire which include the fact that Constantinople, the capital of the Byzantine Empire, was protected with walls that lasted almost the entire 1000 years, making Constantinople, and the center of the empire's economy/culture impervious to invaders.

Athena (goddess of wisdom) gave perseus an unlikely tool to defeat medusa that requires brain over brawn that was instrumental in his victory against medusa

It was a combination of conquests and alliances