Once one studies algebra, one finds categories such as $R-\textbf{Mod}$, abelian groups, sheaves over abelian groups, $\mathcal R-\textbf{Mod}$ and the like. They are all abelian.
On the other hand, it turns out that quite some functional-analytic categories are not abelian.
Most proofs in homological algebra can be done in these "easy" categories using explicit methods, without appealing to category theory. Moreover, it takes quite some effort to prove everything in the categorical framework.
My question hence is this:
What are some really weird abelian categories?
I'd be particularly interested in those where the explicit methods are "difficult".
I already found https://mathoverflow.net/questions/112574/cocomplete-but-not-complete-abelian-category