Easy brainstorming?!

A simple example are all integers.

The answer is $\boxed{\text{Yes}}$ . An example of such a set is $S =$ { $-3,-2,-1,1,2,3$ }.

Edit: I suspect that any such set must have at least 6 elements, as in my example, while from Dr. Bretscher's example we see it can be as large as all of $\mathbb{Z}$ . Any such set must contain both positive and negative values, for otherwise the least positive and greatest negative values could not be the sum of two other elements. Also, there must be at least 3 positive and 3 negative values in order to assure that the largest positive and least negative values can be expressed as sums of two other elements. Finally, the set $S$ need not be "symmetric"; for example, we could add either of the elements $4$ or $5$ to my example set and still have a set that meets the requirements.