Here's a somewhat general form of obstruction: many nice categorical properties imply that a category's classifying space 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 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 $\Lambda$ 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 $BS^1$.

• ...