Phineas Nicolson wrote: "If space has positive curvature, it is a closed universe."
Correct. Positive curvature (a closed universe) means $\rm k>0$ and $\rm \Omega_k<0$ since $\rm k=1/r_k^2=-H^2 \Omega_k/c^2$ where $\rm r_k$ is the curvature radius of the hypothetical 4D hypersphere whos surface is our spatial 3D universe.
The Antipode (the farthest possible proper distance, or in other words the half roundtrip distance) is at a proper distance of $\rm \pi \ r_k$.
Phineas Nicolson wrote: "In spherical geometry, two straight lines meet at two points."
In a radiation dominated Big Crunch universe two photons emitted in opposite directions at the Big Bang will meet each other at the Antipode when the Big Crunch occurs (you never see the back of your own head).
In a matter dominated one they will make one full roundtrip and the photon emitted in one direction at the Big Bang will reach you from the opposite direction at the Big Crunch. In that scenario the two photons meet twice, the first time at the Antipode and finally back where they started (not only in terms of the proper but also in terms of the comoving distance).
If the universe is both closed and static you can always see the back of your own head with a time delay of $\rm \Delta t=2 \pi \ r_k/c$, here the two photons will intercept periodically with an intervall of $\rm \Delta t/2$ (given your head is transparent or you move it out of the photon's way in time).
Phineas Nicolson wrote: "So if space were elliptic, both spaceships would meet at some point not on Earth, in fact it'd be the point 'opposite' from Earth in space, the furthest from it in any direction."
That is the case in the spherical universe. In standard cosmology we have spherical, flat and hyperbolic, due to the Copernical principle the elliptic reduces to spherical since the curvature is isotropic.
