Take as $D$ a complementary $\mathbb{Q}$-vector space of $\mathrm{span}(\{c(n): n\in\mathbb{N}\})$ in $\mathbb{R}$ as $\mathbb{Q}$ vector space. This is not measurable, of course (countably many translate of it covers $\mathbb{R}$ and as you said it can't be of positive measure). It's a version of the Vitali set.

Choose your favorite c(n), and define $D$ as any complementary $\mathbb{Q}$-vector space of $\mathrm{span}(\{c(n): n\in\mathbb{N}\})$ in $\mathbb{R}$ as $\mathbb{Q}$ vector space. This $D$ is not measurable, of course (countably many translates of it cover $\mathbb{R}$, so it can't be of measure zero, and on the other hand as you said it can't be of positive measure). It's a version of the construction of the Vitali set (actually the standard Vitali set already works with a sequence of rationals c(n) ).