What kind of literary experience can we gain from Chekhov's short story The Darling?
A literary experience for a reader is what is called Reader-Response Criticism. This criticism relies upon how a reader relates to a text, how they interpret a text, and the impact the text has upon them (if any). The Reader-Response Criticism relies only on the reader and forgoes the author's intent for the piece. In essence, the reader has an active role in the text given it depends upon their engagement in order to explicate their literary experience.
Given that your question elicits a subjective answer, the one I will provide you with will speak to my own personal reading experience with Chekhov's short story The Darling.
Based upon the fact that Olga's life only has meaning for her when she is with a man, Olga can only identify with herself when married or in a relationship. Therefore, what I come to find is that the lives of many women today are not much different from those in 1899 (when the story was published). Given that I teach high school, I see this happen with many of my students. They, in essence, are identified by their relationships and the boy/young man with whom they are involved.
I find the story very sad. It weighs very heavily upon me knowing that some women lack the internal strength to identify themselves apart from their partner. I feel sorry for women like Olga given I support strong women. The story simply speaks to the weakness seen in women over time and how their oppression is understandable given their weaknesses regarding the opposite sex.