Semigroupoid, Small Category, and Groupoid have group like structures, see https://en.wikipedia.org/wiki/Monoid#Relation_to_category_theory
But there Semigroupoid, Small Category, and Groupoid do not require totality - which I believe that means they do not need to be closed.
How can this be possible that their algebraic structure do not need to be closed? but still have a consistent algebra?
You're right that categories and groupoids aren't total in the usual sense of being able to apply a binary operation to any pair of elements. It's not always possible to compose two morphisms and get a third morphism. Instead, the two morphisms have to match up in a specific way.
Instead of thinking of the composition operation as not being total, it's probably better to think in terms of how this "matching" works.
Categories and groupoids have two parts: the objects and the morphisms, and even if their given by just the morphisms, it's easy to reconstruct the objects as the identity morphisms. So we may as well use both in the description.
Each morphism has a domain object and a codomain object. Two morphisms are composable precisely if the codomain of one matches up with the domain of the other. The result of composing the two morphism has the same codomain as the first morphism and the same domain as the second.
Another way to think of this is that rather than one big set of morphisms, you have small sets of morphisms $\hom(x, y)$ for each pair of objects $x$ and $y$. The morphisms in $\hom(x, y)$ are all of the morphisms with domain $x$ and codomain $y$. Then the composition operation is simply a collection of maps $\hom(y, z) \times \hom(x, y) \to \hom(x, z)$ for any objects $x$, $y$ and $z$.
For me, this way of thinking made groupoids click for me. The description via a single set of potentially composable objects had too many moving parts for me to understand.