Look for the most obvious clues as to when the story was set. Look in the inside front cover, if it is a novel, or a collection of short stories. Look at the copyright page. You should be able to see the dates of publication, including the original year of publication, even if the book has had several reprints. Investigate the author. Do an online search of the name and find out the author's date of birth and death. if appropriate and when he wrote the book. This can give you a good idea of when the author was writing.
Look at the story itself for more subtle clues. Some stories refer to historical incidents. For instance, there may be a reference to the abolition of slavery or to women fighting for the vote. Look at the language itself for further clues. In Jane Austen's stories you will note a more formal type of English that reflects not only the period, and also the characters' social class.
Take note of the everyday life of the characters, as described by the author. Notice the descriptions of clothing and household decor. You will also find clues in the cars or other forms of transport described. Observe the food the characters eat and the work they do. The changing role of women in fiction, for instance, reflects actual change that has occurred in the real world.
Study the language itself. Look for slang and colloquialisms that an author may have used himself while writing the book a century ago. Mark Twain's vocabulary reflected what Americans themselves spoke when he wrote his books.