I too late realised that I had posed the question wrong
What I meant was that the subset should be isolated (which is indeed, dense in no interval). But it appears that this is impossible. I don't know the exact proof, but the trick is that you add the size of the intervals around the elements which include no more elements. This amounts to more than R if the subset is un/over-countable. (shown for example after mapping R onto an interval with arctan - you would get more than the length of the interval).
BTW, The set you described is dense: take an element b, and make an interval around it with length e<1. In the decimal expansion of e/4 (to be safe
), there is a place which is the first that isn't 0. Change one of the decimals of b after that place, and you have another element in the interval. So your set is dense in R.