Well don't all want that! First let's address the obvious: Is your rifle capable of shooting sub MOA and what the heck is MOA? 

MOA is Minute of Angle (1/60 or a degree) or in math terms amounts to a circle at a certain distance. Most folks agree that the MOA at 100 yards is a 1" circle. If your rifle can get three bullets (a group) into a 1" circle at 100 yards it would be considered a sub-MOA rifle. Now can the person shooting that rifle get three shots into that 1" circle? Maybe?

MOA is a constant so as you get farther from the target the circle gets bigger. So at 200 yards the MOA is 2" (approximately). At 300 yards MOA is 3" and so on.

So how do I know if my rifle can shoot sub-MOA? This is going to take some work. Your rifle is going have to setup for the best possible condition to send that bullet down range with minimum disruption. Taking the human factor out is where you start. You will need a sled or rifle vise to hold your rifle on bench so that it won't move prior to firing. It should also be able to make a repeatable shot so you can judge accuracy with a three shot group.

Like all endeavors you have to start with quality components. If you are building your own you need high quality barrel. I would use an 18" or even a 24" Black Hole Weaponry barrel. It has polygonal rifling which is generally thought to be better. I would use a free floated handguard which is also thought as better for accuracy. Use high quality ammunition and that should get you started. Of course this is only the mechanical part of the rifle. The parts that will make it able to shoot sub-MOA.

The human elements will take away from the best rifle and best specs. The human needs quality optics, a good trigger and a steady platform to shoot from.

You can buy a high end rifle that states it can shoot sub-MOA and they might even send a test target somewhat proving it. You should be able to clamp that rifle is a vice a duplicate those shots. Of course you would have to same ammo, the same range, the same conditions (temperature, humidity...), you see where this is going.

So can your AR15 be a sniper rifle? Well that wasn't Eugene's original intent but with a little work, money and practice it shoot those gratifying sub-MOA's.