Unbounded flat Euclidean spaces can be either infinite (e.g., an infinite plane) or finite--e.g., a flat torus, constructed by starting with a square and identifying opposite edges.
Meanwhile, the most obvious example of a space with constant positive curvature is the sphere, which is finite. And there doesn't seem to be an obvious way to cut a piece out and tile it like you could to go from a closed torus to an open plane. So, are constant-positive-curvature spaces inherently an unavoidably finite, or is it possible to construct an infinite space with constant positive curvature?
I think the following may be what you want. By the Killing-Hopf Theorem, every complete connected Riemannian manifold of constant positive sectional curvature is isometric to the quotient of a sphere by a group acting freely and properly discontinuously. Quotient spaces of compact spaces are also compact, so it follows that all complete connected Riemannian manifolds of constant positive sectional curvature are compact. There's a nice proof of the Killing-Hopf Theorem in chapter 11 of John M Lee's Riemannian Manifolds: An Introduction to Curvature.