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).