We learn that Miss Brill doesn't really have a great deal to do and that she is essentially alone. Each Sunday, she goes to the park to listen to a band, but she mostly looks forward to eavesdropping on the people around her because this is the closest she gets...
to social interaction:
[...] Miss Brill always looked forward to the conversation. She had become really quite expert, she thought, at listening as though she didn't listen, at sitting in other people's lives just for a minute while they talked round her.
Miss Brill doesn't seem to realize that the interest she takes in other people's conversations and lives suggests that she doesn't have much happening in her own. She does read the newspaper to an old man a few nights a week, but this seems to be the extent of her "social" interaction, and even he typically falls asleep. Further, she looks around her and sees that
Other people sat on the benches and green chairs, but they were nearly always the same, Sunday after Sunday, and [...] there was something funny about nearly all of them. They were odd, silent, nearly all old, and from the way they stared they looked as though they'd just come from dark little rooms or even -- even cupboards!
But, again, she doesn't realize that she, too, is old. Our first clue is that she seems like she is no longer in great health: she feels a tingling in her hands and arms at the beginning of the story, and this is not a normal, healthy sensation. Miss Brill only seems to realize her age when she is made fun of by a young couople. To his girlfriend, the young man calls Miss Brill, a "stupid old thing" and he meanly jokes that everyone wishes she would keep her "silly old mug at home."
Our last clue that Miss Brill is old and lives a fairly uneventful and sad life comes in the final lines of the story. We learn about how she typically stops at the bakery for a slice of cake and how momentous an occasion it is when her cake has an almond in it. Today, she skips the baker's and returns home, to "her room like a cupboard." She'd previously identified those "old" people as looking like they had just come out of "dark little rooms or even -- even cupboards!" In the end, there is some acknowledgement that she has too. Miss Brill seems to, on some level, come to understand her similarity to these people as, "when she put the lid on [the box which holds her fur] she thought she heard something crying." It, of course, cannot be the fox, and so it must be her.