Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.


The category $\mathcal C$ is naturally additive as a subcategory of complexes of abelian groups.



> 
> Question: Can we write down a complete list of **isomorphism classes** (up to translation) of **indecomposable objects** of $\mathcal C$?
> 
> 
> 


It is easy to see that the number of such isomorphism classes is countably infinite.


Here are some indecomposable objects:
$$
\cdots\to 0\to\mathbb Z\overset{m}{\to}\mathbb Z\to\mathbb Z\_m\to 0\to\cdots,
$$


$$
\cdots\to 0\to\mathbb Z\_{(m,n)}\to\mathbb Z\_n\overset{m}{\to}\mathbb Z\_n\to\mathbb Z \_{(m,n)}\to 0\to\cdots,
$$
where $m$ is a natural number, $n$ is a prime power and $(m,n)$ denotes the greatest common divisor.


But there is more; for instance, if $p$ is prime then the indecomposable object
$$
\cdots\to0\to\mathbb Z\_p\to\mathbb Z\_{p^2}\overset{p}{\to}\mathbb Z\_{p^2}\overset{p}{\to}\cdots\overset{p}{\to}\mathbb Z\_{p^2}\overset{p}{\to}\mathbb Z\_{p^2}\to\mathbb Z\_p\to 0\to\cdots
$$
can have any finite "length."


If this classification problem has been solved, a reference would be great. Otherwise I would very much appreciate any idea/hint towards a general solution.


(I've added the noncommutative-algebra tag because chain complexes can be considered as modules over a certain non-commutative ring. The question I am asking is a sub-problem of classifying all finitely generated indecomposables for this ring.)



Classification of long exact sequences
In a series of recent papers, Schmidmeier and Ringel show than the classification of monomorphisms in the category of finitely generated $\mathbb Z/p^n$-modules is a wild problem of representation theory for $n>6$. Hence, your problem is also wild. There's no hope to get what you want.


