I'm not sure this is on topic for this thread, but I'm posting it here rather than start a new one for something so closely related.

The following quote is from Wiki.

“Dirac hypothesized that what we think of as the "vacuum" is actually the state in which all the negative-energy states are filled, and none of the positive-energy states. Therefore, if we want to introduce a single electron we would have to put it in a positive-energy state, as all the negative-energy states are occupied.”

Does this imply that the vacuum cannot be infinite?

There never was nothing.