Creating a 3D Looping Map

Hello,

I am attempting to make an infinitely-repeating first-person 3D map similar to a 3D implementation of the game Yume Nikki. The map is extremely simple; basically a flat plane with a roof and a couple of static objects scattered around. Once you move to an edge of the map, you find yourself walking in on the opposite side of the map, probably by teleporting the player to the appropriate spot. The edges of the map are not recognizable to the player, it just looks like one endless plane.

My initial idea was to set walls around the map which are projecting camera views of the other sides of the map, kind of like surrounding the map with mirrors. However, I think it's impractical to set the cameras in a way which seamlessly displays the other side of the map. I could also make duplicates of the map surrounding the navigable area, but since the map is purely flat, I think you would ostensibly be able to see an indeterminate number of repeated maps away from the origin (once again, like setting two mirrors opposite to each other). It would be computationally impractical to render that many copies of the same scene. Is there any tool I'm not thinking of that would make this easier to implement?

Thanks in advance!