I know that the Cantor Set is uncountable (this is a well-known result), so I know that there must be something wrong with the following method for counting its elements, but I'm not sure where the flaw is.
I'll assume that the Cantor set in question is a subset of the interval $[0,1] \subseteq \mathbb{R}$ obtained through iterative elimination of middle thirds.
The point with coordinate $0$ is numbered $1$, and the point with coordinate $1$ is numbered $2$. After the first iteration, the middle third of the unit interval is deleted, and we find two more points that must be in the Cantor set. These are the points with coordinates $1/3$ and $2/3$. Number them $3$ and $4$, respectively.
After the next iteration, there are four new endpoints of line segments that must now be counted. Number the points with coordinates $1/9$, $2/9$, $7/9$ and $8/9$ as $5$, $6$, $7$, and $8$, as depicted below.
Every point in the unit interval must either not be in the Cantor set, or be the endpoint of a line segment at some finite iteration of the deletion process. But every such endpoint is numbered by the process above, so every point in the Cantor set should correspond to a non-negative integer, and vice versa.
I think the problem is that there exist points that are in the Cantor set that are not endpoints of a line segment at some iteration, and so would not be counted. However, I can't think of an example of such a point ...
I thought about this while musing over the following questions:
Is it possible to draw an uncountably infinite number of non-intersecting circles in the plane? (clearly yes, e.g. concentric circles with radii in $[0,1]$)
Is it possible to draw an uncountably infinite number of non-intersecting figure eights in the plane? (I think the answer is no, but I am not sure why)
While thinking about the second of these, I considered a figure eight with two figure eights inside of it, one in each half, and two inside each of the halves of those, and so on:
These figure eights are easily countable. Number the biggest one $1$, the second biggest ones $2$ and $3$, and so on. This is closely similar to how the points in the Cantor set are "counted" in the first part of this question.
I'm dealing with two sets. One is the Cantor set; the other is the infinite set of nested figure eights, the first few of which are in the diagram above. The latter is clearly countable, while the former is ostensibly not. What is the difference between the two?
Is there possibly some way to construct a set of figure eights with centers at points in the Cantor set so that these figure eights don't intersect? Would this be a way to draw an uncountable infinity of figure eights in the plane? If not, is there another way?
Here is one example. Start with the whole interval, $[0,1]$. In the first step, go to the left sub-interval, $[0/3,1/3]$. In the second step, go to the right interval, $[2/9,3/9]$. Continue this process, alternating between left and right child, an infinite number of times. Choosing any element from the interval at each step, you obtain a sequence of numbers which converges. And since $\mathbb R$ contains the limits of all convergent series, that sequence corresponds to a real number. On the other hand, that real number can not be an endpoint at any level of the iteration, because at every finite level, the corresponding sequence element will be a whole interval, and two steps later both endpoints of that interval will have been cut away. Therefore you have a number which is in the Cantor set but not an endpoint. Changing the pattern of how you alternate between left and right children will result in other examples as well.
If you want to be more explicit, the limit of the series described above satisfies the condition $x=\frac19x+\frac29$ so you can conclude that $x=\frac14$. So in fact this is the number that ruler501 mentioned in a comment to your question, although I didn't realize this when I wrote my answer.
Regarding the second part of your question, about the figure-eight, you should probably follow the post mentioned by Gerry Myerson, Countable or uncountable set 8 signs. I must confess that dealing with the first part of your question, I myself would have imagined this to be possible, similar to the Cantor set construction but using areas at each step which thin out to curves in the limit. Although the mentioned post has me convinced, I still don't see exactly where such an approach would fail.