Theorem: Given $A$ a boolean ring/boolean algebra then there is an equivalence of categories between the category of $A$-modules and the category of sheaves of $\mathbb{F}_2$-vector spaces on Spec $A$. The equivalence sends every sheaf $\mathcal{M}$ of $\mathbb{F}_2$-vector space to its space of section, $\Gamma(\mathcal{M})$ which is a module over $\Gamma(\mathbb{F}_2) = A$.
Proof: Spec $A$ has a basis of clopen given by the elements of $A$. Using Grothendieck comparison lemma, this allows to give a more algebraic description of sheaves on Spec $A$ as:
For each $a \in A$ a set $F(a)$, with (functorial) restriction maps) $F(a) \to F(b)$ when $b \leqslant a$ such that the natural maps:
$$ F(a \cup b) \to F(a) \times F(b) $$
is an isomorphism when $a \cap b = 0$.
This allows to exhibit the inverse construction: Given a $A$-module $M$, we define
$$ \mathcal{M}(a) = aM $$
with the restriction map $aM \to bM$ being given by multiplication by $b$. One easily check that if $a \cap b = 0$ then $aM \times bM \simeq (a \cup b) M$ hence $\mathcal{M}$ is a sheaf of $\mathbb{F}_2$-vector space such that $\Gamma(\mathcal{M}) = \mathcal{M}(1) = M$.
Conversely, starting from any sheaf of $\mathbb{F}_2$-vector spaces $\mathcal{M}$ we have $\mathcal{M}(1) = \Gamma(\mathcal{M}) = \mathcal{M}(a) \times \mathcal{M}(\neg a)$, and the action of $a \in A$ on $\Gamma(\mathcal{M})$ is the identity on $\mathcal{M}(a)$ and zero on $\mathcal{M}(\neg a)$. It follows that $a\Gamma(\mathcal{M}) = \mathcal{M}(a)$, and from there we easily check that the two constructions are inverse of each others.