In this answer, mme shows that for any compact Lie group $G$, there is a model for its classifying space $BG$ which is the direct limit of closed manifolds. If $G$ is discrete (i.e. $\dim G = 0$), then a model for $BG$ is also a model for $K(G, 1)$ since $\pi_i(BG) \cong \pi_{i-1}(G)$.

Let $G$ be a countable abelian group and $n > 1$. Does there exist a model for $K(G, n)$ which is a direct limit of closed manifolds?

Recall, a space $X$ is a model for $K(G, n)$ if $\pi_n(X) \cong G$ and $\pi_i(X) = 0$ for $i \neq n$.

There is such a model for $K(\mathbb{Z}, 2)$, namely $\mathbb{CP}^{\infty}$ which arises as the direct limit of finite-dimensional complex projective spaces $\{\mathbb{CP}^m\}_{m\geq 1}$. This can also be viewed in the context of the linked answer as a model for $BU(1)$ is a model for $K(\mathbb{Z}, 2)$.

The above question may be too difficult to answer in general. I would be interested to know what happens for the next simplest cases, namely $K(\mathbb{Z}/m\mathbb{Z}, 2)$ and $K(\mathbb{Z}, 3)$.