One of the biggest issues with Emily was her father. He was of the "old south" where men were chivalrous and proud. He would not let any men in to "court" his daughter because he felt that she was too good for all of them. No man was good enough for his daughter, so she was isolated that way. This means that she had no interaction with any outside people in the town. She knew no one. She conversed with people only when her father had people over. She was very isolated.
When he died, she didn't know how to handle it. She had never been away from him or his "rule." In fact, she wouldn't let the coroner take his body away until after a couple days went by. Then the first man to ever show any interest in her was Homer Barron. He never meant to stay around, so she did what it took to keep him "around" forever.
There are several social changes that occured within Emily's life.
The society that once witnessed old-fashioned courtships, kinships, old family names, an economy by means of hand labor, and the perennial presence of the Southern way of life is, in the story, now steadily shifting towards industralism, capitalism , social expansion, racial integration, and the view of the South as anachronistic.
Emily is there, caught in between the two clashing worlds, without the protection of her "almighty" father, and with no ties that are strong enough to help her make the transition towards the present.
The social issue in itself is the inevitable reality of change and shifting towards the future by severing the only ties you know. This is a universal problem that happens in every generation, and needs the willingness of each individual in it to make the effort and embrace change.