How would Special Relativity work if the universe were periodic?

The technical term for this sort of universe is a torus. It is possible for a three torus to be flat.

In such a universe you can have a set of observers that are all stationary with respect to each other, this is the same as in a standard flat universe, however, in a torus observer A is to the left of observer B who is to the left of observer ... Z who is to the left of observer A.

Now, just as in our universe, you can have multiple sets of such observers, say the ones described and another set A’, B’, ... Z’. All of the primed observers are at rest to each other and all of them measure the unprimed observers as having the same speed. Locally everything follows the standard rules of SR.

However, there is one big difference: if the unprimed observers measure the distance between A and B and add it to the distance between B and ... add it to the distance between Z and A they will get a number for the size of the universe. If the primed observers do the same procedure they will get a different number for the size of the universe. If many different sets of observers do this they will find that there is one unique set of observers for whom the size of the universe is maximum.

So although such a universe behaves locally like we expect, globally it violates the first postulate. There is a uniquely identifiable reference frame that can be singled out by physical measurements.

The time that passes on your clock between event E and event F is given by the length (in the Minkowski metric) of the path you followed to get from E to F. You can compute this length by integrating the spacetime interval along this path.

If I remain on earth while you head outward along a path that eventually brings you back to earth again without accelerating, we can take E to be the event of your departure and F to be the event of your return. You and I both traveled from E to F but along different paths. Those paths have different lengths, and therefore our clocks can show different amounts of time passed. Exactly what that difference is depends on the details of the spacetime geometry and the path you followed.