jmc

Any number in $S$ has a leftmost (eights) digit that is equal to $1$. The remaining three digits can either be $0$ or $1$, so there are a total of $2^3=8$ elements in $S$. Notice that element $x$ in $S$ can be paired with another element $10111_2-x$, which is the base $2$ number whose right-most three digits are the opposite of those of $x$. Thus, the sum of the elements in $S$ is equal to $4\times 10111_2=100_2\times 10111_2={\boxed{1011100}}_2$.