The story begins with the narrator's recognition that "Miss Emily Grierson died," and the speaker says that she is like a "fallen monument" in their community. This certainly sets a somewhat somber and even melancholy mood, as the death of someone who feels like a "monument" in the community must shake it to its foundations. In the second paragraph, words like "obliterated," "stubborn," "decay," and "eyesore" continue this mood. They even add a sense of deterioration and decline that accompanies the idea of society sort of figuratively crumbling and the actual appearance of the buildings literally moldering. Words that describe Miss Emily—"tradition, [...] duty, and [...] care" as well as "hereditary obligation"—help us to understand, even before she is described in more detail, how old-fashioned she is and what a drain she can be on her town, economically and socially. People sort of tiptoe around her as a result of their sense of "obligation" and this increases her mysteriousness too.
In all, there is a mood of cheerlessness and dreariness—of descent, and decline, and rot—and this is all incredibly appropriate given what we learn as the story progresses. Miss Emily, of course, does experience a sharp mental decline after her father's death, leading her to poison her lover, Homer Barron, in order to prevent him from abandoning her.