Expanding on my comments, here are some obstructions coming from Hausdorff dimension and self-similarity. 

1. One observation is that every nonelementary Kleinian group $\Gamma$ has positive **critical exponent** $\delta$.  Furthermore, $\delta$ equals the Hausdorff dimension of the **conical limit set** of $\Gamma$, $\Lambda^c$, the subset of the limit set $\Lambda$ consisting of conical limit points. In particular, limit set cannot have zero Hausdorff dimension. Now, take a Cantor subset $C\subset \mathbb R$ which has zero Hausdorff dimension (see e.g. [here](https://math.stackexchange.com/questions/73547/uncountable-sets-of-hausdorff-dimension-zero) for a construction). Then $C$ cannot be the limit set of a Fuchsian group. 

2. Take two Schottky groups $\Gamma_1, \Gamma_2< PSL(2,\mathbb R)$ with disjoint limit sets $\Lambda_i, i=1,2$, that have different Hausdorff dimension. Then $C=\Lambda_1\cup \Lambda_2$ cannot be a limit set of a Fuchsian group (for instance, because of self-similarity of a limit set: It has the same "local" hausdorff dimension everywhere). 

With more thought, one can surely get other examples. 

On the other hand, every nonempty compact subset $C\subset \mathbb R$ can be realized the Hausdorff-limit of a sequence of limit sets of Fuchsian groups $\Gamma_n$ (i.e. $\lim_{n\to\infty} d_{Haus}(C, \Lambda(\Gamma_n))=0$). 

For proofs of basic facts about Hausdorff dimension of limit sets and critical exponents, see for instance 

<cite authors="Nicholls, Peter J.">_Nicholls, Peter J._, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, 143. Cambridge etc.: Cambridge University Press. xi, 221 p. \textsterling 19.50; {$} 34.50 (1989). [ZBL0674.58001](https://zbmath.org/?q=an:0674.58001).</cite>  

It is a bit dated, but, still the best textbook on this subject.