Solution:

The answer is $|S| \leq 8$. An example is $S=\{n \pm \sqrt{2} \mid n=1,2,3,4\}$ (and any of its subsets).

In general, construct a graph with vertex set $S$ in which we join two numbers with rational sum. We claim this graph is bipartite; indeed if $a_{1}+a_{2}, a_{2}+a_{3}, \ldots, a_{n}+a_{1}$ are all rational for some odd $n$, solving the resulting system of equations gives $a_{1}$, $\ldots$, $a_{n}$ all rational numbers.

Accordingly we may 2-color $S$. If $|S| \geq 9$, then we may find a set of five numbers with rational sum, as desired.