The problem with Sand Jack's `proof' is that it improperly uses mathematical induction. We have
  1. The length of figure 0 (Sand Jack's first figure) is 2.
  2. If the length of figure k is 2, then the length of figure k+1 is 2.
From this, mathematical induction allows us to establish that the length of figure n is 2, for any natural number n. However, induction does not allow us to make such a claim for the limit. Sand Jack's proof is parallel to the following, also incorrect, one:
Consider the sequence 1, 1/2, . . ., 1/n, . . . . Clearly, each element of this sequence is greater than zero. Hence the limit of this sequence (that is, zero) is greater than zero. Therefore, zero is greater than itself.

I have heard Sand Jack's proof attacked with ``the limit of the sequence of figures is not a straight line''. However, the limit is a straight line, for a reasonable definition of `limit': for each figure n, let f_n be the natural parametrisation of that figure. Then the sequence { f_n } converges (uniformly) to the natural parametrisation of the diagonal line.

To reiterate: after any finite number of steps, the length is indeed 2, but the figure is not a straight line. In the limit, the figure does become a straight line (not a fractal as Dhericean supposes), but the length is not 2. Dhericean and evan927 state the first part of this, but the second is just as important.