To expand on Niels' answer: given a connected semilocally simply connected topological space $X$ and a point $x\in X$, the fiber functor $F_x:\mathrm{Cov}_X\rightarrow\mathrm{Set}$ from the category of covering spaces of X yields an equivalence of categories $F_x:\mathrm{Cov}_X\xrightarrow{\sim}\pi_1(X,x)\mathrm{Set}$ with permutation representations of the fundamental group $\pi_1(X,x)$ by sending a covering space $\pi:Y\rightarrow X$ to the permutation representation of $\pi_1(X,x)$ on the fiber $\pi^{-1}(x)$ induced by monodromy. However, $\mathrm{Cov}_X$ is very much not a Galois category: according to SGA1 or Stacks, a Galois category is equivalent to the category of finite (continuous) permutation representations of a profinite group, whereas $\mathrm{Cov}_X$ is equivalent to the category of possibly infinite permutation representations of a (discrete) group. I suppose $\mathrm{Cov}_X$ could be called an infinite Galois category, although this conflicts somewhat with the notion of infinite Galois category used to define the pro-étale fundamental group.

On the other hand if you want to work with an honest Galois category: given a connected topological space $X$ and a point $x\in X$, the fiber functor the fiber functor $F_x:\mathrm{FCov}_X\rightarrow\mathrm{Fin}$ from the category of finite covering spaces of X yields an equivalence of categories $F_x:\mathrm{FCov}_X\xrightarrow{\sim}\widehat{\pi}_1(X,x)\mathrm{Fin}$ with finite (continuous) permutation representations of the profinite completion $\widehat{\pi}_1(X,x)$ of the fundamental group. Note that we do not need to assume that $X$ is semilocally simply connected for this to work: in the presence of a universal cover of $X$ the fundamental group is equivalently the automorphism group of the universal cover, but in general the (profinite) fundamental group can be recovered as the automorphism group of the fiber functor.

Why would one want to consider only finite covers of topological spaces? One reason is to formulate comparison theorems: for instance given a scheme $X$ (locally) of finite type over $\mathbb{C}$ and a point $x\in X$ we have an equivalence of Galois categories $\mathrm{FEt}_X\xrightarrow{\sim}\mathrm{FCov}_{X(\mathbb{C})}$, but this is obviously false at the level of infinite covering spaces.