You mentioned string theory, so you may know that compact dimensions with a small enough radius don't measurably alter the world's long-scale physics (otherwise string theory would already be falsified!). This is basically what happens with gravitational physics in the universe you propose. The result is that there is no gravity at all.
First, to build an intuition for what spaces like this would look like to an inside observer, you can download the app Curved Spaces. If you look at the "Torus Cubic" space within the "Flat" folder (keep the left arrow key pushed to remove the walls), you will see that any object seems to have infinite repeated copies across all three dimensions. Your proposed universe is like that, but objects only repeat across one dimension (up/down).
Now for the physics. Unless I'm missing something, five meters is a length scale big enough for electromagnetism, the strong and weak forces, and quantum mechanics in general to develop similarly to our 3D world; this would mean that hydrogen and helium atoms would probably form as usual.
However, the relevant length scale for gravity is much larger (think about the size of stars and planets), so gravitationally speaking your world would approximately behave like a 2D world. And it is known that general relativity in two dimensions is a purely topological theory without propagating degrees of freedom: modulo technical details, spacetime is always flat in this world.
This means that gravity as we know it wouldn't exist: no planets or stars would form, you would just have a primordial gas of particles without a force to condense them.
If you still want to impose the existence of stars and planets by hand, a moment of thought will convince you that they have to be cylindrically symmetric to the inside observer. My guess is that their external behavior would be that of the corresponding black hole-like solution, a so-called open cosmic string. As explained in that link under the "Gravitation" section, open cosmic strings don't produce a gravitational force either, only topological effects. So you would still have no orbits, the artificially-added cylindrical planets and stars would just move at constant speed without interacting unless they collide. If you are interested, this paper describes the interior solution of such a body and other possible configurations.