I can think of three reasons:
First, he didn’t make a lot of films. He only directed 13 movies, and two of those were low-budget “B” movies that were… well, not great. So, when you do 11 films over 43 years, that’s not a lot of chances to win.
Second, he didn’t lobby for the award. Oscars are won after a fierce campaign season, when agencies and studios spend millions of dollars on ads for their clients and films… While Warner Brothers certainly promoted both Kubrick and his films, he didn’t care for it (or LA, for that matter). Here’s a friend describing him: “Social standing means nothing to him and he has no interest in acquiring it; money serves exclusively to guarantee him independence.” He shunned the glamor of LA, moved to England, and never looks back.
Third, his films rarely aligned with the zeitgeist the way competing films did. He did films that was ahead of their time, like Dr. Strangelove (he lost to the director of My Fair Lady); scandalous, like A Clockwork Orange (he lost to William Friedkin, who directed The French Connection); or was challenging and distant like Barry Lyndon (which lost to Milos Foreman’s work on One Flew Over the Cuckoo’s Nest). The only movie that he directed that was a cultural phenomenon, 2001, lost to Carol Reed and Oliver (of all films), and I suspect the fact that is was SF was a huge factor in that loss. The stars never aligned, and so… no Oscars.