What does it really mean when we say that the probability of something is zero? Conventionally, people will say a probability of zero is equivalent as saying that the event is impossible.

But when we look at the probability from a mathematics perspective, probability is defined as the frequency of occurrence over the how many times the experiment is performed, limit as the number of trials goes to infinity. 

Doesn't this mean a probability of zero is an occurrence that is arbitrarily small but possible? What are some of the ways to make this line of argument more rigorous?
One way of making this rigorous is to make an analogue between probability and area. The comparison here is made precise via measure theory, but can be explained without recourse to technical definitions.

Consider two "shapes": a point and the empty shape consisting of no points. Think of these geometric entities as sitting inside two-dimensional space. What is the area of a point? Zero. What is the area of nothing? Also zero. Does that mean that a point is the same as nothing? Of course not. All we can say is that the notion of area is not capable of distinguishing between them. 

Using the above example, we can construct two games. In the first game, take a black dartboard and paint a single point red. In the second game, leave the dartboard black. Throw a dart at either one. Is the probability of hitting the red different between these games? Does this mean that these games are equivalent? Does this thought experiment suggest anything about the limitations of the probabilistic method?
If probability is the number of occurrences, then a probability of zero means there will be zero occurrences.  Zero doesn't signify an infinitely small number of occurrences, it signifies no occurrences at all.

Sometimes we don't need an infinite number of experiments to determine that the frequency of occurrence will always be zero.  Something as simple as solving a system of linear equations can illustrate the point.  If the system is inconsistent then it will have no solution set. You can experiment by plugging in values all the way to infinity and there will never be a solution set.

Whatever the problem, if it can be defined as a system of linear equations and it then solves to an inconsistent system, the solution set is empty.  Parallel lines will never cross.  In that case the probability is zero.  

So probability of zero doesn't mean an infinitely small number of occurrences, it means absolutely no occurrences at all.