Anyone who asked you this question, using the word “stuffy” to describe it, is not an “informed reader.” In other words, the asker does not quite understand what literature is.
First, this play is far from stuffy in terms of its action—three sisters, of diverse social character, and with interesting occupations, sharing a mutual backstory, still mourning a lost father, confronted with a real social and economic dilemma, hounded by military leeches, visited by a fascinating stranger—this is not a superficial drama in which to imbed some philosophical point of view—this is a real dramatic scenario, a real “conflict” to be resolved. Secondly, Chekhov’s work “matters” in the same sense that dramatic history “matters”—as educated people, we are interested in how and why artistic expression changes through history, and these “comedies” are ideal representatives of how drama moved from artificiality to reality in the late 19th-early 20th century. Thirdly, studying any social situation, carefully, enlightens us on the dynamics of all social “contracts”—this could be any family in the modern age solving similar dilemmas, similar restricted choices—for example, what are the universal dynamics of moving out from your father’s house and protection, or how does one balance one’s career choices with one’s family life, or how do the genders exist socially by the unwritten, unspoken rules of social gender expectations, or what is the universal appeal of big cities to country folk? We might as well ask “What does the Sermon on the Mount, that stuffy old story, matter in today’s society?” Finally, literature has no debt to future “societies”—its raison d’etre is much more complex than “to matter in tomorrow’s society.” Its aim is to matter on the personal, aesthetic level.