Let $A$ be a subset of $\mathbb{R}^2$ which intersects every straight line in exactly two points.
Is there a such set which is Lebesgue measurable or Borel?
A well-known fact is that there exists such set which is not Lebesgue measurable.



Is there a set that intersects every line twice which is Lebesgue measurable or Borel?
There is such a set which is Lebesgue measurable, and indeed of Lebesgue measure zero. To see this, start with a subset $S$ of $\mathbb R^2$ such that every line intersects it in continuum many points, for instance $C\times\mathbb R\cup\mathbb R\times C$, where $C$ is the Cantor set. Now repeat your favorite transfinite recursive construction of a set $A$ intersecting each line at exactly two points, modifying it in such a way that we all the points picked belong to $S$. Since $A$ is a subset of a measure zero set $S$, it itself is Lebesgue measurable of measure zero.


It appears that existence of such a set which is Borel is an open problem, see [this MO post](https://mathoverflow.net/a/21864/30186).


