Why aren't things massless when they have balanced forces acting on them? Since $0$ divided by anything is $0$, why isn't everything with a net force of $0$ massless? Since $F = ma$ rearranges to $m = F/a$, when $F = 0$, $m$ should equal $0$, shouldn't it? This obviously can't be true, because it doesn't make any sense; but why?
 A: When you have $F=0$ all you can say is that $m a=0$.
Now, in classical mechanics, the mass $m$ is a property of the object you are studying, and it is generally assumed that $m > 0$. Thus you can divide $ma=0$ on both sides by $m$ and obtain $a=0$, which gives you Newton's first law: an object with no net force applied to it, moves at constant velocity.   
See that you cannot divide both sides of $ma=0$ by $a$ because $a=0$ and division by $0$ is not defined.
The reason we take $m>0$ in classical mechanics is as follows. Classical mechanics lets you calculate the movement of objects using your knowledge of the forces applied to it and the equation $F=ma$. You see that if you were to put $m=0$ in the equation, you need either $F=0$ (there can be no force whatsoever) or $a=\infty$ (things fly around at infinite velocity, whatever that means) or other combinations in which the theory would be completely useless.
So classical mechanics $F=ma$ is a useful theory only when considering objects which do have a mass. Now, there are objects with no mass (like the photons: the particles of light), but for those you need other theories.
A: You've uncovered an additional assumption of classical mechanics!
Classical mechanics is not just Newton's laws. The equation $F = ma$ contains two new, unknown quantities: the force $F$ and the mass $m$. (We already know how to measure acceleration.) As a result, the equation is totally useless unless we also know something about $F$ and something about $m$. 
The information from $F$ comes from force laws, like Hooke's $F = -kx$ or gravity $F = mg$. However, even knowing what force is, the equation remains useless because it can always be satisfied if you allow mass $m$ to vary in a really weird way. 
Newton knew this, which is why he addresses this in the very first sentence in Principia Mathematica! He gives a totally useless definition of mass, but one that makes it clear that the mass of an object doesn't change unless the object itself is changed. (He essentially says "Okay, we all already know what mass is. It's just the amount of stuff.") That's why you don't get $m = 0$ when $a = 0$. It's assumed to stay the same.
A: I agree with the other answers, but consider also the following.
Physical reality is modeled by mathematics, but mathematics is not physical reality. If one considers causality as a constraint to physical reality in the frame of classical physics, then there is a better way to express Newton's  2nd law by considering the logical order in which things occur
$$a=\frac{\Sigma F}{m}$$
since it's the net force that leads to movement, not the movement that leads to force.
Not that $\Sigma F=ma$ is wrong but rather incomplete, since cauaslity was not taken into account.
A: For F=ma when F=0, mathematically either m=0 or a=0 or both. But we know that m is not zero (unless stated that the body is massless), which gives a=0, where a is the acceleration of the center of mass of the system.
If you manipulate the equation as m = F/a, it doesn't make any difference as if we know that the body isn't massless, the acceleration (of COM) should be zero and your equation will yield 0/0 which is indeterminate form and is not equal to zero.
