Let's stipulate that
Connective -- i.e. nonnegatively-(homologically)-graded -- chain complexes have a very natural geometric interpretation: by the Dold-Kan theorem, they are a way of thinking about simplicial abelian groups.
Dually, coconnective -- i.e. nonpositively-graded -- chain complexes have a very natural geometric interpretation as complexes of functions on spaces.
The motivations for considering the category of all (unbounded) chain complexes, then, are quite good -- this category provides a home for both the connective and coconnective chain complexes, and has excellent formal properties like stability and a good duality theory.
However, these motivations are quite formal in nature -- they don't provide a geometric interpretation in line with (1) or (2) above. For the most part, these motivations operate "one category level higher", discussing properties of the category of chain complexes. I'm specifically looking for something which gives a geometric, natural way to think about an individual chain complex.
Question: What is a good geometric interpretation of nonconnective, noncoconnective chain complexes?
Notes:
A similar discussion would more generally stipulate that grouplike $E_\infty$-spaces have a natural geometric interpretation, and ask for a similarly "geometric" interpretation of more general spectra. I'd be equally happy with a discussion in this setting.
Similarly, I'd be happy with a discussion in the context of complexes of sheaves of various flavors.