The term "sport" has such a broad definition that it can be applied to just about every game or competition out there. If you think paintball is a sport then good for you, but other people may not be so inclined.
So, by these qualifications, paintball is a sport. It's team-based, has a governing body and is fairly wide-spread.
But the game you made up that one summer called "Bleep-Bloop" and six people know about it is not.