This claim is somewhat true, but misleading. NASA can't protect us if a major asteroid (like the one that created the Chicxclub crater and hit the dinosaurs) hits us, but that's not because of a lack of workforce. If a major asteroid hit earth (greater than 50m wide), it would probably wipe out Earth completely. There would be acid rain, debris everywhere, and shockwaves. However, most of the asteroids that hit earth (which is more frequent than people might think) are relatively small and have little impact. Furthermore, tiny asteroids (like most that hit earth) only get bright enough to be detected in the last few hours before they hit Earth. Also, for larger asteroids, NASA has the sentry detection system and regularly tracks large asteroids and their orbits so they can determine if there is a threat to hitting Earth way in advance. Therefore, the only asteroids that NASA can predict only a few hours before impact are teeny ones, which pose little to no threat.
https://www.nasa.gov/solar-system/nasa-system-predicts-impact-of-small-asteroid/
