The zombie apocalypse sounds outlandish, but there are many examples in nature of disease, parasites or even fungus profoundly changing behavior in a zombie-like way. Take, for example, the aggressive effects caused by rabies or the so-called zombie ants whose behavior is changed by an infecting fungus in a way that benefits the fungus just before it kills the host ant. The video game The Last of Us explored how such a similar scenario might realistically occur with humans.
If technology develops to the point where minds can be controlled, then a zombie apocalypse is possible. If this technology falls into the wrong hands, such as that of a ruthless dictator, he can use it to control all of his subjects into mindless zombies who will do his bidding.
A zombie apocalypse is not possible, because zombies, as portrayed by the media, do not exist. There may be occasionally zombie like behaviors in people, such as the actions of individuals who become addicted to bath salts and the occasional involuntary movements of those who suffer from certain brain injuries and disease. However, there is nothing with the ability to cause large swathes of individuals to become undead, brain-eating monsters.
Zombies are the undead that are brought back to life and have an insatiable hunger for brains. The only case in nature where zombies are a reality are certain ants in Brazil which are plagued with a certain fungus. Even still, these zombie ants do not run around attacking other ants and attempt to eat their brains, they just move the ant's body to a more suitable location for fungus growth and then the ant stops moving again. A much more likely scenario is destruction by mass hysteria as a result of food shortages due to natural disasters, not some sort of made-up zombie apocalypse.