  The category ${\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient
$B_{k,n}$ of a preprojective algebra provides a categorification of the cluster
algebra structure on the coordinate ring of the Grassmannian variety of
$k$-dimensional subspaces in $\mathbb C^n$, \cite{JKS16}. Among the
indecomposable modules in this category are the rank $1$ modules which are in
bijection with $k$-subsets of $\{1,2,\dots,n\}$, and their explicit
construction has been given by Jensen, King and Su. These are the building
blocks of the category as any module in ${\rm CM}(B_{k,n}) $ can be filtered by
them. In this paper we give an explicit construction of rank 2 modules. With
this, we give all indecomposable rank 2 modules in the cases when $k=3$ and
$k=4$. In particular, we cover the tame cases and go beyond them. We also
characterise the modules among them which are uniquely determined by their
filtrations. For $k\ge 4$, we exhibit infinite families of non-isomorphic rank
2 modules having the same filtration.
