Question

Why did the classical civilizations seem to need radical social inequalities?

Answer 1

Higher class needed to suppress lower class in order to keep them "at bay".

Answer 2

The classical civilizations depended on an underclass of laborers (often in the form of slaves) to support the higher classes that were creating the "classical" advancements of civilization in government, philosophy, the arts, etc.

Philosophers in the classical era justified these social inequalities.  The Greek philosopher Plato proposed that the ideal state, described in his writing, The Republic, would have three classes of persons in it.  At the bottom level are the producers--farmers, craftsman, laborers--who make the basic things society needs.  At the top level are the guardians or rulers, whom Plato envisioned to be the wisest members of society, sort of philosopher-kings.  In between there would also be a class of auxiliaries, the spirited type of men who would serve as military protectors of the state.  Plato's model envisioned people fit into the class within society for which they were most naturally suited.  

Aristotle also (the next great classical era philosopher in Greece) believed that human beings were naturally predisposed to certain types of life.  He justified the institution of slavery by saying that some persons were just better suited by their nature to be commanded by other and be in subservient roles, for the overall good of society.