Stable order on binary strings
No. Given any linear ordering of the $n$-bit strings, find a sequence of at most $n+1$ strings going from the first string in the ordering to the last string in the ordering by single-bit changes per step. Some two adjacent strings $x$ and $y$ in this sequence must be at least $2^n/n$ steps apart from each other in the ordering. But then, unless your definition of "far" only allows linearly many strings to be far apart from each other, there must be some $z$ that is far from $x$ and $y$ but between them in the ordering. So changing one bit from $x$ to $y$ reverses the ordering with $z$.