I think one of the reaons why this is such an excellent story is precisely because it is timeless. It represents the transforming ability of love by showing us how Gurov's relationship with Anna turns him into a completely different person at the end of the story from the casual seducer that we meet at the beginning. Gurov changes from regarding women as "the lower race" to recognising in Anna a true soul mate that he feels linked to.

Chekov's short story addresses the human characteristics of loneliness, desire to be loved, a propensity for longing, and a seeming inability to be satisfied.  As the two characters exchange small talk, the "lady" remarks:  "Time goes fast, and yet it is so dull here!" Perhaps this is an affliction of the middle and upper classes...the poor are too busy struggling to survive to worry about the problems of boredom.  Yet, many people can relate.  Chasing the next dollar, the next woman, has been a problem forever.

The conclusion of the story is also universally identifiable.  In the story's final lines,  Dmitri tries to console himself: 

There's something pathetic about her, anyway," he thought, and fell asleep.

Remember Aesop's fable of the fox who was unable to reach the luscious looking grapes?  He decides that the grapes were probably sour anyway.  Who hasn't tried to convince themselves, in one way or another, that what we thought we wanted was really no good?  Therein lies the universality. 

A link to the full text of the story appears below, as does a link to further analysis here at eNotes.