This is a complex question to answer. On the one hand, the story ends with the suicide of its protagonist. In addition, arguably, Edna kills herself because she had no other option. When Robert abandoned her because of the way he was unable to break free from society and its restrictions in the same way that she had done, it was clear she had little hope for the future. She would not be able to return to her meek and submissive role of mother and wife, and therefore she was running out of options. Death, and a return to the embrace of the "abysses of solitude" of the sea seems the only option. It is important perhaps to point out that in the last paragraphs of the final chapter, when the seductive voice of the sea is described once again, it is accompanied by the following image:
A bird with a broken wing was beating the air above, reeling, fluttering, circling disabled down, down to the water.
Given the image of the caged parrot at the beginning of the text and how this represented Edna's position, this bird image at the end of the text could likewise symbolically represent Edna's inevitable death. Thus it is possible to regard this text as the failure of a quest.
However, it is important to remember a conversation that Edna has with Doctor Mandelet in Chapter 38 when she comments on the process of awakening and the joys that it brings the person who sets out to become truly self-aware:
The years that are gone seem like dreams—if one might go on sleeping and dreaming—but to wake up and find—oh! well! Perhaps it is better to wake up after all, even to suffer, rather than to remain a dupe to illusions all one’s life.
Edna here states that the year she had of living her own life her own way, aware of herself and being in control of her life. Even though this period only lasted for a year, it is much better than a life time of living one's life "sleeping and dreaming," playing out a submissive state. Whatever the "suffering" experienced, Edna feels her process of awakening was most definitely a victory of the soul rather than a failure, in spite of the ending. Better to have savoured life in its fullness for even a brief time than to come to the end of one's life and never having lived.