It is a well known result of Sierpinski that there exists non-measurable subsets of $\mathbb{R}^{2}$ which intersect each line in at most two points.

1. W. Sierpi´nski: Sur un probl`eme concernant les ensembles mesurables superficiellement. Fund. Math. 1 (1920), 112–115.

It is a well known result of Sierpinski that there exists non-measurable subsets of $\mathbb{R}^{2}$ which intersect each line in at most two points. Furthermore, there exists a real-valued function whose graph is a non-measurable subset of $\mathbb{R}^{2}$.

1. W. Sierpi´nski: Sur un probl`eme concernant les ensembles mesurables superficiellement. Fund. Math. 1 (1920), 112–115.

2. Gelbaum, Bernard R., and John M. H. Olmsted. Counterexamples in Analysis. San Francisco: Holden-Day, 1964. (p. 142-145)