The category of $C^*$-algebras is not abelian (a "proof" that it is pre-abelian can be found here, but it does not seem correct; I can't find any authoritative sources). However, it's possible to do K-theory over the category of $C^*$-algebras, and one can often directly use tricks and techniques from homological algebra, especially when working with exact sequences. Sometimes people even use directly definitions and terms in homological algebra that are not really well-defined if the underlying category is not abelian.

The question is, why does this "just work"? Is it that the category of $C^*$-algebras have some good properties that allow one to use techniques from homological algebra, or is it that a lot of homological algebra don't really require the category one is working in to be abelian (e.g., pre-abelian is actually enough)?

I apologize if this question is not research-level enough.