Simply stating a distance a paintball will go isn't the easiest thing to do as the real answer is that it depends. This is because there are issues of effective range, safe range and absolute range to consider.
The first issue is the effective range of a paintball. Paintballs have a balancing act that they must achieve: they must be hard enough that they fly free of a barrel without breaking, while also being fragile enough that they can break when they hit their target. The result of this balancing act is that paintballs will not break unless they are moving at a certain speed. In effect, this means that if you shoot paintballs a very long distance, near the end of their trajectory they will slow to the point that they will not break even if they hit their target. If you go to the back of a paintball field you will almost inevitably find unbroken paintballs that people lobbed too far and they simply fell to the ground, unbroken. The effective range of a paintball gun depends on the paint and what your opponents are wearing (softer clothing does stop paintballs from breaking), but is usually around 80-100 feet.
The next issue is the safe range. Paintball guns need to be chronographed to a safe shooting velocity. If paintballs are shot at too high of a velocity, they can injure someone when they hit, so most fields have a cap on the rate of speed that paintball guns can shoot, commonly at around 280 feet per second (FPS) or 200 miles per hour (MPH). If you fire paintballs at this velocity and angle them up and try to lob them as far as possible, the maximum range will be around 100 yards.
Now, let's assume that you aren't concerned with hitting someone too hard or worried about the paintball breaking, and instead just want to shoot a paintball as far as you possibly can. To do this, you would find the hardest paintball available that is the least likely to break and you would crank up the velocity on your gun so that it shoots as fast as possible. Theoretically, if you continued to find harder and harder paintballs and continually increased the velocity, you could shoot an indefinite distance. In practice, though, this will not work as paintball guns generally have a maximum FPS that they will ever be able to shoot unless you significantly modify the gun. With this variability on maximum shooting speed, each gun will have a different maximum absolute range, though some guns may be able to shoot as far as 150 yards.
Since paintballs, no matter what gun they are shot from, are all governed by the same laws of physics, different guns will not shoot paintball guns farther and different barrels will not lead to further shots if they are shooting the same speed, unless they change something about how the paintball is shot. The two things that can change about how a paintball is shot at a given velocity are the rotation of the paintball and the shape of the paintball. Specialized equipment can modify both of these things.
The first way to increase shooting distance is to change the rotation of the ball. For firearms, such as rifles, accuracy and distance are improved by rotating the ball perpendicularly to the ground by rifling the barrel which is simply putting grooves in the barrel that force the bullet to spin. Paintball manufacturers have experimented with similar rifling, but this has proven ineffective because deep enough grooves to spin the ball will also lead to the paintball breaking in the barrel (which is not an issue with a firearm). Paintball manufacturers have created the Flatline (Compare Prices) and Apex (Compare Prices) barrels which put horizontal rotation on the paintball. Specifically, by giving the paintball back spin they are able to increase the distance the gun can shoot. Note, though, that this does nothing to increase the effective range of the gun: you may be able to shoot further, but you will still likely have the paintball bounce, rather than break, if it is further than 100 feet.
The second approach to increasing shooting distance is to change the shape of the paintball itself. First Strike rounds do this by shaping them more like a bullet with fins that direct and spin the paintball as it flies through the air. Since the paintball will also shoot only one way through the barrel (nose first, fins at the rear), engineers also were able to make the nose more brittle, so the paintball will have more durable sides (so it won't break in the barrel) and a brittle nose (so it will break on target); this actually increases the effective shooting range. This, though, does come at some significant weaknesses. First strike rounds must be fed into a gun in the proper direction meaning that you will need to use magazines to feed the rounds which hold significantly fewer shots. Plus, the price of these rounds is much higher than standard paintballs and can cost close to a dollar a round (Compare Prices). First Strike rounds should not be used with a Flatline or Apex barrels. The effective range of these may be as far as 200 feet and the absolute range may reach 200 yards.