Don't worry about it as paracompact manifolds have all the topological properties of metric spaces. In particular, they are perfectly normal Hausdorff spaces. Manifolds are also commonly required to be second-countable. This is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space. However, non-Euclidean Hausdorff spaces do exist (probability approaches certainty) as the existence of manifolds in Hausdorff spaces includes non-Euclidean manifolds.