Let's start with a look at what tragedy is . . . Aristotle defined it as “an imitation of an action that is serious, complete, and of a certain magnitude; in language embellished with each kind of artistic ornament, the several kinds being found in separate parts of the play; in the form of action, not of narrative; through pity and fear effecting the proper purgation of these emotions.”
O'Neill certainly includes all of these characteristics in his autobiographical play. Since the play is based on his own life, it is very clearly a realistic imitation of an action. There is great magnitude of the several subplots - the mother's addiction, the men's alcoholism, Edmond's consumption, etc. The language is absolutely poetic throughout - consider Edmond's monologue about the sea as a perfect example of artistic expression. The play is full of emotion - emotion that is purged throughout the four acts. Each character has at least one major cathartic moment. Because of the realism and the seriousness of the story-line, an audience may also experience a catharsis.
What makes it modern, I suppose, are the modern issues driving the tragedy, along with the more domestic (less epic than traditional tragedies) setting.
However, O'Neill did not set out to write a tragedy. He set out to heal himself by facing the demons of his youth. The play was written not to be published or performed, but was given to his wife as an anniversary gift; he credited her with bringing him into the light from his dark past. She had the play published after his death as she recognized its literary merits.