In the present paper the author introduces the concept of almost slender modules, which is very useful for studying the Cartesian products of modules over a Dedekind domain. The ring R is called slender [see \textit{E. L. Lady}, Pac. J. Math. 49, 397-406 (1973; Zbl 0274.16015)] if for every homomorphism \(f: \prod^{\infty}_{i=1}A_ i\to R\), where \(\{A_ i| i=1,2,...\}\) are R-modules, there is \(n\in {\mathbb{N}}\) with the property f(\(\prod^{\infty}_{i=n}A_ i)=0\). Let R be a slender Dedekind domain with a countable set of nonzero ideals M. The R-module G is said to be almost slender if G does not contain nonbounded cotorsion R-modules and R-modules which are isomorphic to the module of Baer- Specker P over the ring R (i.e. \(=\prod_{\aleph_ 0}R)\). In the main result of the paper the author characterizes the almost slender R- modules. Theorem. The R-module G is almost slender if and only if for every set of R-modules \(\{G_ i| i\in I\}\), where \(| I|\) is a cardinal less than the first cardinal of measure not zero, and for every homomorphism \(\phi\) : \(\prod_{i\in I}G_ i\to G\) there is a finite subset J of the set I such that the R-module \(\phi\) (\(\prod_{i\in I\setminus J}G_ i)\) is bounded.