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 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

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.

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

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 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 the 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

Nicholls, Peter J., The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, 143. Cambridge etc.: Cambridge University Press. xi, 221 p.   (1989). ZBL0674.58001.

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