It should be a cantor-set. Also, by construction, the length is preserved in each step, so the Haussdorff dimension should be 1.

To prove that it is in $[0,1]$, depends on how you construct your sets.
If you impose the restriction that the segments in each step is within $(0,1)$, it should be trivial.

I assume you look for a general, fractal and self-similar construction.

I think you'll run into trouble by moving it half the size (it will self-intersect, I believe), but if you move it the full size, it is fine. This is easy via induction (make a picture).

EDIT: Here is a picture of my construction, each step on its own line.
The picture is a bit deceptive in the end, it really should be "gray" or something.

[![fractal lines][1]][1]


  [1]: https://i.sstatic.net/GG5tO.png