Yes, to add emphasis, there is no Dirac Sea. This was an early idea that turned out to be incorrect. They were not antiparticles, they were supposed to be normal particles in negative energy states, and antiparticles were the vacancies ("holes"). But Quantum Field Theory replaced these negative energy states with positive energy antiparticles. (And no, they do not travel backwards in time!)
The idea of having states fully occupied up to a certain energy, the Fermi level, survives in condensed matter scenarios, e.g. conductors and neutron stars, but negative energy states don't occur in Quantum Field Theory.