  Let $\mathcal{M}$ be the moduli space of rank 2 stable torsion free sheaves
with Chern classes $c_i$ on a smooth 3-fold $X$. When $X$ is toric with torus
$T$, we describe the $T$-fixed locus of the moduli space. Connected components
of $\mathcal{M}^T$ with constant reflexive hulls are isomorphic to products of
$\mathbb{P}^1$. We mainly consider such connected components, which typically
arise for any $c_1$, "low values" of $c_2$, and arbitrary $c_3$.
  In the classical part of the paper, we introduce a new type of combinatorics
called double box configurations, which can be used to compute the generating
function $\mathsf{Z}(q)$ of topological Euler characteristics of $\mathcal{M}$
(summing over all $c_3$). The combinatorics is solved using the double dimer
model in a companion paper. This leads to explicit formulae for $\mathsf{Z}(q)$
involving the MacMahon function.
  In the virtual part of the paper, we define Donaldson-Thomas type invariants
of toric Calabi-Yau 3-folds by virtual localization. The contribution to the
invariant of an individual connected component of the $T$-fixed locus is in
general not equal to its signed Euler characteristic due to $T$-fixed
obstructions. Nevertheless, the generating function of all invariants is given
by $\mathsf{Z}(q)$ up to signs.
