Does every l.e.s. “in homology” come from a s.e.s. of complexes?

Given a long exact sequence of the form $$\dots\to A'_n \to B'_n \to C'_n \,\xrightarrow{\omega_n}\, A'_{n-1} \to B'_{n-1} \to C'_{n-1}\to \dots\qquad (*)$$ is there a way to recover a short exact sequence of complexes $\mathcal A=\{A_n,\partial_n^A\}$, $\mathcal B=\{B_n,\partial_n^B\}$, $\mathcal C=\{C_n,\partial_n^C\}$ such that the sequence (*) "is" the long exact sequence in homology induced by $$0\to \mathcal A\to \mathcal B\to \mathcal C \to 0$$ and the morphisms $\omega_n$ are in fact the connection morphisms of that homology? I mean $A'_n\cong H_n(\mathcal A)$ for all $n\ge 0$ and similarly for $B'_n$, $C'_n$.

I expect the answer will be "obviously no", but then is there a case in which it is possible?

-