What is it about Miss Celia's background that makes her different from the other club woman?
Simply put, Miss Celia is not cut from the cloth of supposed perfection from which the other club women see themselves cut. Miss Celia was born into a poor family and does not see herself as worthy or needing all that is around her. Due to her own background as an "outsider," she does not fit into the inclusive world of the ladies at the club. She is not able to be assimilated into the world of the club, a clique that seeks only to be exclusive. Miss Celia further complicates this relationship with the club as she married Hilly's former boyfriend.
The ultimate reason why she cannot join the club is because of her inability to conceive. While no one at the club forbids her from joining the club because of this, her inability to conceive is a part of the narrative that shows her to be imperfect and an "outsider." Miss Celia's condition as an outsider makes her able to embrace Minnie and people of color, in general. Her background as someone who knows what it is like to be relegated to the outside makes her able to appreciate what it is people of color are enduring. She experiences it herself, showing that the practice of exclusion was intrinsic to segregation.