If I understand the question correctly, a subset $l$ of lines belongs to $S$ if and only if there is a straight-line segment crossing exactly the lines from $l$.

The problem is then equivalent to the following dual version:

What is the maximum VC-dimension of a finite set of points in the plane, with respect to the double-wedges that do not contain the origin?

Where by "double-wedge" I mean the union of two opposite regions of the plane between two crossing lines.

It is easy to see that a set of $6$ points in convex position cannot be shattered. Since every set of $17$ points in convex position contains a convex $6$-gon, the VC-dimension is at most 16.

The next step could be investigating all configurations of $6$ points plus the origin if there is some configuration that can be shattered.

[The answer was edited after Gjergji's comment.]

One can, indeed, show that the VC-dimension is at most $10$ by a counting argument outlined by Gjergji.
A pair of faces containing the endpoints of the segment determines the subset $l$. For lines in general position, there are $${1+{n+1 \choose 2} \choose 2}$$
pairs of faces, which is still more than $2^n-1$ for $n=11$. But some of the subsets $l$ were overcounted: every $1$-element set was counted $n-1$ times and every $2$-element set at least twice, so we can subtract $3/2 \cdot n(n-1)$. In this way, we get an upper bound $2046$ for the number of subsets $l$ for $n=11$, which is just enough to show that the VC-dimension is at most $10$. Further improvements are possible, for example by considering overcounted triples or by considering the faces with more than $3$ vertices (using
this result).

**Edit:** According to P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry, Chapter 8.2 or MathSciNet, MR1274574,
Harbort and Moller [1] proved that every simple arrangement of $9$ pseudolines in the projective plane contains a subarrangement of six pseudolines with a hexagonal face. No such arrangement can be shattered: a triple of pseudolines determined by every other edge of the hexagonal face cannot be crossed by a pseudosegment that avoids the other three pseudolines. This shows that the VC-dimension is at most $8$, even in the stronger setting of pseudolines in the projective plane.

[1]: H. Harbort and M. Moller, Esther Klein problem in projective plane, J. Combin. Math. Combin. Comput. 15 (1994), 171--179.