A climax is a moment in the story where the conflict or tension reaches its highest point. It is a stage in the narrative that all the action builds up to. At times, it is also referred to as the crisis. The crisis is preceded by what is known as the rising action. The climax is usually followed by the falling action which would include a resolution or denouement that signifies the end of the story.
To determine the climax in "Eveline," one has to ask which particular moment all the action in the story builds up to. It is evident that Eveline's deepest desire is to be free from the restrictive and suffocating life of drudgery that she has been living most of her life. All of her thoughts and most of her actions revolve around the idea that Frank will take her away from her miserable situation. She even goes as far as accompanying him to the station to board a ship to Buenos Aires. The climax of the story is reached when she finally decides not to leave with Frank. Her decision is depicted in the following extract:
No! No! No! It was impossible. Her hands clutched the iron in frenzy. Amid the seas she sent a cry of anguish.
What follows is the falling action. Frank calls to her but she refuses to acknowledge him. Her mind has been made up. She has resolved the crisis in her heart and in her mind by deciding to remain where she is. Eveline has, ironically, chosen to continue living an unfulfilled and miserable life.
The short story "Eveline" written by James Joyce is a part of Dubliners. This is early-Joyce and the plot-structures are still more or less traditional and coherent and hence the question of a climax. Such Freytagian categories would become untenable in his late work.
The climax in "Eveline" is a complicated matter. The story uses the stock fairy-tale plotline of an incarcerated beauty and the romantic promise of a journey of liberation coming for her lover, Frank in this case. A classic climax of such a plot is the lovers' successful elope and it is this very structure that is subverted carefully by James Joyce. Eveline, who is often seen as a spirit of Ireland, is undecided even at the end and in the final final moment of departure, she remains stock-still and denies Frank. I think this moment is the climax of the story when Eveline does not go with Frank and the ship starts to move toward Argentina. The quick resolution is marked by Frank's incomprehension of Eveline's facial expression. The climactic moment of the story is thus made up of a subversive twist.