Is it possible to achieve total equality within a society? Why or why not?

No, it is not possible to achieve total equality in a society.

While the Declaration of Independence declares that "all men are created equal," it does not mean that everyone is equal in intelligence, physical skills, etc. It means that they hold equal rights that are inherent in all people. This concept is one propounded during the Enlightenment of the eighteenth century, an age of great thinkers such as Jean-Jacques Rousseau and John Locke. These Enlightenment thinkers believed in the concept of natural rights, rights that allow people to freely make their own choices and have the opportunity to prosper.

This Enlightenment concept of natural rights--"all men are created equal"--is greatly different from the "equality" established by the 211th, 212th, and 213th Amendments to the Constitution and enforced by agents of the United States Handicapper General Diane Clampers. In order to make Harrison Bergeron "equal," the young man must wear handicaps that weigh him down and disguise his good looks, and shocks are sent to his brain.

People are made differently; it is impossible for them to be the same unless artificial controls are used, such as the noises sent through the brain and the heavy handicaps that are worn. And, when equality is enforced, the best that can be hoped for is equal mediocrity.
George Bergeron is highly intelligent, and so, he must wear forty-seven pounds of birdshot in his vest and headphones that emit sound to distract him in order to make him "equal."

"If you could just take a few [the bird shot] out when you came home from work," said Hazel. "I mean-you don't compete with anybody around here. You just sit around."

His having been made to be "equal" has obviously depressed George as he "just sits around." So, in a sense, his rights to be fulfilled and happy have been taken from him. His forced mediocrity has taken the spirit from George. His son Harrison actually rebels, but then is killed for his actions by the Handicapper Diane Glampers. 

Equality is impossible to establish without enforcement, but with such enforced equality, talents are hampered and intelligence not allowed to be exercised, and, therefore, people are dissastified, unhappy, held back and, therefore, unequal.