The answer to this question is bound to be subjective (based on opinion), but I would say the answer is yes. The novel's story remains relatable and realistic today.
The realism of the novel, in my opinion, grows from its continuing "relatability". The main themes of the novel dealing with family, identity, and problems of faith are perennial and will probably always be relatable.
As these themes anchor the story and essentially function as the meaning of the novel, offering the reader a sense that even if the characters are fictional, the problems they face are drawn from the real world.
The novel is set around WWII, but changes in society and technology since then have not eliminated Antonio's issues - being challenged to please both of his parents, challenged to "find himself", and challenged to determine his spiritual beliefs.
Of course, there are elements of the story that seem less realistic. Antonio's dreams and Ultima's powers are not fully believable to all readers, but this was as true at the novel was written as it is today.