This is a take on 1.C.7 of Axler's "Linear Algebra Done Right", but changed to "the smallest nonempty subset" instead of "an example of a nonempty subset".
When I was first doing this problem I thought of the subset ${(0,0),(1,1)}$ where $(1,1)+(1,1)=(0,0)$. This trick makes sense to me after reading the Digression on Fields section in Chapter 1. However, I am not sure how to prove that this subset is the smallest that satisfies the requirements or if my answer violates some rule of math.
The solutions manual gives $Z^2$ as the answer because it fails the scalar multiplication requirement but had consistent addition and additive inverses.
There is no smallest: consider $ ... 2k \mathbb{Z} \times \{0\} \subsetneq ... \subsetneq 4 \mathbb{Z} \times \{0\} \subsetneq 2 \mathbb{Z} \times \{0\} \subsetneq \mathbb{Z} \times \{0\}$. None of these sets is even minimal.
A smallest subset of $\mathbb{R}$ closed under addition and "minus" must be contained by every one of these sets, so it's contained by their intersection $S_0 = \{(0, 0)\}$. Since you're looking for nonempty suitably closed sets, the smallest has to equal $S_0$. However, that's a subspace.