I suppose one of the central aspects of this story that makes it timeless is the way that, although it is obviously based in one context, it contains elements and themes that are universal and are more about the human condition rather than anything else. This means that even for us, who are separated from the actual time of writing the story by so many years, can identify these elements and themes, and because we can relate to them and apply them to our lives, the story maintains its relevance and timeless quality.
There are so many to choose from, but for me I would have to pick the ending as being the perfect demonstration of this. Note how throughout the story, the narrator, the governess, is actually presented as an unreliable narrator. We never know if she really can see the ghosts that she tells us about, or if she is making them up, psychologically creating a reason for her employer, the attractive ward of Miles and Flora, to come and rescue her. It is clear from their first meeting that she is deeply impacted by his character, but that she lacks enough perspicacity to realise her own feelings. Note what happens at the end to continue this theme of ambiguity as the governess and Miles are on the bed together and Peter Quint is supposedly haunting them:
I caught him, yes, I held him--it may be imagined with what a passion; but at the end of a minute I began to feel what it truly was that I held. We were alone with the quiet day, and his little heart, dispossessed, had stopped.
From her point of view, the governess believes that Miles has died out of the shock of being haunted, but the way in which she holds him with such a "passion" suggests that she actually might have suffocated him. The ambiguity of the writing means we never actually know for sure, but are left with these doubts. A timeless classic.