A nice way to think about this type of thing is to imagine a bunch of $\infty\times\infty \times 5$ ft blocks stacked next to each other, with the same stuff in each block. So if you were in one block, looking into another, you would see another copy of yourself and the room you are in, and another one beyond that, and so on, repeating infinitely. In math, this is called a covering space. I will refer to the strange space you describe as cylindrical space.
Something interesting about this space is that it is not simply connected. Imagine you had a string in normal space. You could tie the ends together, and then do whatever you wanted with it. You could wad it up and put it in your pocket, for instance. But in this space, you could tie one end of your string in one of the blocks to the other end in the next block, and you would suddenly have an infinite strand which you could not wad up without breaking! Moving down from the covering space to cylindrical space, it would look like a an infinitely long piece of string. This is a really weird concept which is not true in normal space.
Everything, obviously, would have to fit in this very narrow space. There is really no way around this. Every room only has two walls (unless you want it to be even narrower), so all the electricity and plumbing has to run to these two walls, or through the floor. Also, all things we normally put on walls, like TVs, paintings, mirrors, etc,... would have to go on these two very narrow walls. An architect would have to take advantage of vertical space to make lots of rooms. Beds would have to lie in the infinite direction because nobody taller than $5$ ft could lie down in the $5$ ft direction. Nevertheless, skinny houses and buildings are possible. Something like this narrow house might be what you get.
It's not all bad. Even though the room is narrow, you don't have to worry about everybody seeing the TV; there are a whole array of TVs, one in each copy of the universe all stacked next to each other. Since you only need two walls, the building will be stronger. Also, you don't need so many mirrors in the bathroom because it is really easy to see the back of your head. There would be more advantages, I think if the universe were a little wider. Say, $30$ ft. Then you could take advantage of the cylindrical properties to have hallways that wrap around. Then, instead of two rooms being at opposite ends of the hall $30$ ft apart, the furthest apart they can be is $15$ ft. The same advantage applies to lots of application where minimizing distance is an issue.
Some things could seem infinite, like beds or bathtubs five feet wide. That would be pretty cool.
All in all, this would be a pretty weird place to live.