On this particular video thread the comments are very inquisitve and entertaining.  Here is one of the longer and more in depth ones:

One of the commenters had this to say:

The physic answer is that, in the currently accepted view of nature, yes you are allowed to consider infinite number of steps and this is tackled in field theory (likelihood of all possible particle paths have to be weighted against each other), BUT you are not going to be able to do an observation that exhibits the infinity possible divisions of a segment. What matters to physics is what you are capable to actually measure, and you are going to be always limited by the very physical world (Heissenberg’s uncertainty principle for instance makes it clear within the realm of quantum mechanics) and your instruments: the size of 2^n becomes smaller than the precision of any physical measurement instrument’s accuracy as n grows large enough. One “solution” of the paradox is that the protagonists of the tales are actually point-like beings with zero size. If you account for the space occupied by the objects, the paradox vanishes.

That you are able to draw a line of length sqrt(2) was not so obvious to the pythagoreans. They thought that we humans can merely depict a distorted representation of the “true mathematical world”, so although the drawn line might appear correct, there is no certain way to know for sure if its length is exactly sqrt(2). I think that they were thinking that if you were able to draw a straight line to represent all the numbers, it should be “filled”, but they couldn’t cope with the notion of the “holes” in the line (the irrational numbers). The modern solution, of course, if that we can show (or state) that the set of real numbers posses the “completeness property” I think this deserves a video!

We hope you like this video and look forward to more cool ones from Numberphile.  This guy really knows a lot about numbers!