Must a subset of $\mathbb{R} ^n$, that has a dimension of $1$, be formed by a union of lines?

Question

Let $S$ be a subset of $\mathbb{R} ^n$ with a box-dimension of $1$. Is $S$ necessarily a union of lines (including "curved" lines and line segments)?

If no, then is $S$ at least an infinite set of points that can be totally disconnected, such that $S$ doesn't form a line?

Thanks