If $l>k$ then the required condition always holds, so we assume that $1\le l\le k$. Let $n=|S|$. Then there are ${n\choose l}$ subsets of size $l$ of $S$, and each of the given $r$ subsets contains ${k\choose l}$ of subsets of size $l$. Therefore $r\cdot {k\choose l}\le {n\choose l}$, which provides a lower bound for $n$.
Now let us try to provide a more refined analysis.
Let us call a quadruple $(n,k,l,r)$ of natural numbers with $l\le k\le n$ admissible, if there exists a family of sets satisfying the condition from your question.
Your question admits two interpretations via studied subjects.
The first interpretation belongs to graph theory. According to it, of natural numbers with $l\le k\le n$ is admissible iff the generalized Kneser graph $K(n,k,l-1)$ has a clique of size $r$. In particular, when $l=1$ this holds iff $n\ge rk$.
The second interpretation is combinatorial. Namely, given natural numbers $n\ge d,k$, let $A(n, d, k)$ be the maximal possible number of binary vectors of length $n$, (Hamming) distance at least $d$ apart, and constant weight (that is, the number of $1’$s) $k$. This subject looks more studied than the previous, see the references. According to this interpretation, a quadruple $(n,k,l,r)$ of natural numbers with $l\le k\le n$ is admissible iff $r\le |A(n,2(k-l+1),k)|$.
Now I am reading the first paper, trying to use it to obtain bounds for $n$.
References
[E] Joakim Ekberg, Geometries of Binary Constant Weight Codes, Master thesis, Karlstadt Universitet, Faculty 2 Department of Mathematics, 2006.
[EV] Tuvi Etzion, Alexander Vardy, A New Construction for Constant Weight Codes.