I calculate that the total mass in the universe is infinite.

If there is an infinite number of negative energy states an electron can occupy and all the states must be filled with the lowest energy state being an infinite sea of electrons, then the total mass of the universe must be infinite.

The postulate is outdated and should quickly be forgotten. Dirac attributed the negative energy solutions of the Dirac equation to excited electrons that leave "holes" in the Dirac sea. But we now know that these negative energy solutions are in fact positive energy solutions carried by the antiparticles of the electrons. There is no Dirac sea (although there are some nice condensed matter examples, but these are not fundamental).

But the anti-particles were just unoccupied spaces within the filled negative energy levels that Dirac postulated would act exactly like a particle with opposite charge. The fact that there is proof of anti-particles supports the theory, it does not replace it.