It is estimated that around the time of Julias Caeser, the world population was about 1 million. Today, there are over 1 million people in New York City alone.
By the year 3000, the world population should be in the trillions; possibly jillions. It's amazing that Fry was able to get an apartment-sized closet.
Of course, it's possible that most of the world's population was destroyed in either of the two alien invasions.