Hypnos is a first-person narrative written from the perspective of an unnamed character living in Kent, England. The narrator writes that he fears sleep, and is resolved to write his story down lest it drive him further mad, regardless of what people think after reading it.
The narrator, a sculptor, recounts meeting a mysterious man in a railway station. The moment the man opened his "immense, sunken and widely luminous eyes", the narrator knew that the stranger would become his friend-–"the only friend of one who had never possessed a friend before". In the eyes of the stranger he saw the knowledge of the mysteries he always sought to learn.
From this point on, the man and the narrator begin living together, the one sculpting the other during the day, and at night, exploring worlds beyond human comprehension. Over time, the narrator's teacher begins speaking of using their ability to transcend into the unknown to rule the universe, via a set of drugs, a thought that scares the narrator (who disavows to the reader any such hubris).
Soon the narrator is off on a foray with his friend, travelling through a void that he explains is beyond human sensation. Passing through several barriers, eventually the narrator comes to one he cannot cross, though his friend does. Opening his "physical eyes", the narrator wakes up and awaits the return of his friend, who awakes severely shaken and reticent, warning only that they must avoid sleep at all cost.
From then on, with the aid of drugs, the two avoid sleep, as each time they succumb, they both seem to rapidly age and are plagued by nightmares that the narrator refuses to explain. The story ends with the narrator explaining that one night, his friend fell into a "deep-breathing sleep" and was impossible to arouse. The narrator shrieks and faints, and awakes surrounded by police and neighbours, who inform him he has never had a friend -- all there is in the room is a statue of his friend, engraved with the Greek word 'ΥΠΝΟΣ (Hypnos).