A two-point set cannot be $F_\sigma$, as Mohammad mentions in his question. Also,
A two-point set cannot contain a dense $G_\delta$ subset of an arc.
This was proved by Gareth Davies in his thesis (Oxford, 2011), but I do not think he ever published this result. To my knowledge, no better results are known in the Borel-sets-shouldn't-work direction.
In the other direction, the best known results belong to Arnie Miller. He showed that
If $V=L$ then there is a co-analytic two-point set.
See this paper of his from 1989 where he writes about this question and other such things, like nicely-definable Hamel bases and MAD families. Also see this related paper of Zoltan Vidnyansky's, which appeared more recently and extended/streamlined some of Miller's work.
In an unpublished manuscript that he made available here on his website, Arnie Miller also showed that
It is consistent for the Axiom of Choice to fail badly, but still to have two-point sets (e.g., two-point sets can exist even when there is no well-ordering of the reals).
Ben Chad did quite a bit of work about 5-10 years ago in trying to eliminate the Axiom of Choice from constructions of two-point sets as much as possible. Some of this stuff made it into this paper, joint with Robin Knight and Rolf Suabedissen.