Closed uncountable sets of Hausdorff dimension zero turn up from time to time in practice (one of my papers with Nikita Sidorov features one) but I think that there does not exist a set which is simultaneously famous, closed, uncountable, and of zero Hausdorff dimension. I think that to some extent the sets which we call "fractals" become famous due to having striking visual properties, but zero-dimensional sets are necessarily very sparse, and it is perhaps more difficult for such a sparse set to be picturesque.
Let me extend this answer with a more mathematically substantial observation. The precise meaning of "fractal" is notoriously slippery, but one possible interpretation of this term refers to a set such that every open neighbourhood contains an arbitrarily small conformally-distorted copy of the whole set . I would like to mention the following theorem of Falconer (Dimensions and measures of quasi-self-similar sets, PAMS 1989):
Theorem: Let $(X,d)$ be a compact metric space with Hausdorff dimension $s$. Suppose there exist $a,r_0>0$ such that for every $r$-ball in $X$ with $r<r_0$ there is a function $\phi \colon X \to B$ satisfying $ard(x,y)\leq d(\phi(x),\phi(y))$ for every $x,y \in X$. Then the box-counting dimension of $X$ equals its Hausdorff dimension, and its $s$-dimensional Hausdorff measure is finite.
If a set with this property has Hausdorff dimension zero then since zero-dimensional Hausdorff measure is simply counting measure, the set must be finite. There is therefore no zero-dimensional infinite compact set which is "everywhere locally self-conformal" in the sense of the above theorem. This result generalises Nikita's answer; in particular it excludes any compact set $X$ which satisfies the self-similarity property
where each $f_i$ is a $C^1$ conformal contraction, irrespective of whether or not separation conditions such as the Open Set Condition are satisfied.
 There is a lively literature on sets which contain arbitrarily small affinely-distorted but highly nonconformal copies of themselves, so this definition of "fractal" is in practice too restrictive.