The impression given in the story, though no details are provided, is that Miss Emily's father, Mr. Grierson, was a very controlling man. One thing that contributed to Miss Emily's not finding a suitable husband was her family's social status, or at least the social status the Griersons wanted to project. The narrator states that the people of the town "believed that the Griersons held themselves a little too high for what they really were." Thus Mr. Grierson many have turned away suitors if he didn't think they were of the right social status: "None of the young men were quite good enough."
Second, Mr. Grierson may have simply intimidated young men from calling on Miss Emily. The narrator paints a picture, a "tableau," of Mr. Grierson standing inside the gate of his home, holding a whip, with Emily in the background at the front door of their home. That he is seen "clutching a horsewhip" indicates that others were frightened of him and that he dominated Emily completely.
Third, it is also implied that Emily's own strangeness may have contributed to her lack of prospects. The narrator mentions a great-aunt of Emily's who "went completely crazy." Furthermore, the narrator observes, "even with insanity in the family she wouldn't have turned down all of her chances if they had really materialized." Insanity in the family is one thing, but in Emily herself it could be a deal-breaker. Latent or pending insanity in Emily, perceived by others as an odd personality, could have been part of what kept Emily's marital bliss from materializing.
Although the story doesn't exactly say, the reasons Emily remained single into her thirties may have had to do with her high social class, her intimidating and controlling father, and her mental condition.