Here's a somewhat general form of obstruction: many nice categorical properties imply that a category's [classifying space](https://en.wikipedia.org/wiki/Nerve_(category_theory)) is contractible. For example any category with any of the following structures has a contractible classifying space:

 - an initial or terminal object

 - binary products or coproducts

 - even just _any_ functorial way to embed two objects $X,Y$ into a common object $X \to F(X,Y) \leftarrow Y$

 - ...

[Here's a note](https://www.matem.unam.mx/~omar/notes/contractible.html) by Omar Antolin-Camarena exploring some of these properties.

So if you have a category whose classifying space is not contractible, then chances are it's not very "nice" from a categorical perspective. For example:

 - The category of fields has a disconnected classifying space. So does the category of algebraically closed fields.

 - The category of algebraically closed fields of characteristic $p$ has a classifying space $BGal(k)$ where $k$ is the algebraic closure of $\mathbb F_p$ if $p \neq 0$ and $k = \overline{\mathbb Q}$ if $p=0$.

 - [Connes' cyclic category](https://en.wikipedia.org/wiki/Cyclic_category) $\Lambda$ has classifying space $BS^1 = \mathbb C\mathbb P^\infty$.

 - It follows that $Ind(\Lambda)$, a sort of "category of cyclic sets" also has classifying space $BS^1$.

 - ...