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I would describe it as "outdated". It is formally equivalent to the more modern, quantum field-based description, but I think people don't like the way it poses things conceptually.
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The idea of the Dirac sea was really meant to explain the origin of the positron, which acts just like an electron except it has positive charge. A positron can be created out of nothing so long as an electron is created simultaneously.
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The Dirac Sea idea is that what we call "empty space" is really an infinite sea of electrons with "negative energy". If you hit one of those electrons in the sea very hard, it gets promoted up to positive energy, leaving behind a hole in the sea.
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