Subset of plane with measure $1$ in all lines

Under CH, there is such a set. To construct it, list all lines $\{L_i : i < \omega_1\}$ and construct $S$ inductively as follows. Start with empty $S$. At stage $i$, suppose $S$ is contained in $\bigcup \{L_j : j < i\}$ and it meets every $L_j$ at length one, for $j < i$. Note that $L_i \cap \bigcup \{L_j : j < i\}$ is countable. So we can choose a compact subset contained in $L_i$ of length one which is disjoint with $\bigcup \{L_j : j < i\}$ and add it to $S$. Of course this $S$ is not measurable but Theorem 7 here of Kolountzakis and Papadimitrakis shows it can't be.

Note that this construction also works as long as all sets of size less than continuum are null. But I don't immediately see how to do this in ZFC alone.