*I am still studying relevant sources, trying to provide the required bounds. 
So I am going to update this answer accordingly.*

If $l>k$ or $r=1$ then the required condition always holds, so we assume that $1\le l\le k$ and $r\ge 2$. 
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 $n$. 
The other lower bound, $n\ge\frac{k^2r}{k+(r-1)(l-1)}$, can be easily obtained from [this](https://math.stackexchange.com/questions/4686187/a-family-of-subsets-each-of-size-r-must-witness-decent-size-of-intersection-fo) question.

On the other hand, if I understood the notation in [this](https://math.stackexchange.com/questions/3666339/find-a-family-mathcala-subseteq-n-choose-k-such-that-mathcala)
question right, it suffices to pick $n\ge 2k\max\{k,\sqrt[l]r\}$. 

Let us try to provide a more refined analysis.

We 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, a quadruple $(n,k,l,r)$ of natural numbers with $l\le k\le n$ is admissible iff the [generalized Kneser graph](https://en.wikipedia.org/wiki/Kneser_graph#Related_graphs) $K(n,k,l-1)$ has a clique of size $r$.
In particular, [when $l=1$ this holds iff $n\ge rk$](https://en.wikipedia.org/wiki/Kneser_graph#Cliques).

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 $r\le A(n,2(k-l+1),k)$.


*References*

[E] Joakim Ekberg, *[Geometries of Binary Constant Weight Codes]( https://www.diva-portal.org/smash/get/diva2:6620/FULLTEXT01.pdf)*, Master thesis, Karlstadt Universitet, Faculty 2 Department of Mathematics, 2006.

[EV] Tuvi Etzion, Alexander Vardy, *[A New Construction for Constant Weight Codes](https://arxiv.org/pdf/1004.1503.pdf)*.