Here is a follow-up questions to Must a Borel set B of nonzero measure contain an interval as a subset??
The answer to the original question is NO. For example, $\mathbb{R}\backslash\mathbb{Q}$ or the fat cantor set.
I apply the definition of sets of finite perimeter from Maggi's book.
A Lebesgue measurable set $E$ in $\mathbb{R}^n$ is a set of locally finite perimeter in $\mathbb{R}^n$ if for every compact set $K\subset \mathbb{R}^d$, we have $$\sup\left\{\int_E \mathrm{div} T(x)\mathrm{d}x: T\in C_c^1(\mathbb{R}^n;\mathbb{R}^n),\mathrm{spt} T\subset K, \sup|T|\leqslant 1\right\}<\infty$$
If the above quantity is bounded independent of $K$, we say $E$ is a set of finite perimeter.