Such a measure cannot exist. I am just typing the argument.

Such a measure cannot exist. Suppose to the contrary that we have an uncountable family of lines $\ell$ such that $\mu(\ell)>0$. Then there is $\epsilon>0$ and an infinite family of lines $\{\ell_i\}_i$ with $\mu(\ell_i)\geq\epsilon$. These lines intersect at countably many points. Since the measure has no atoms, this set $E$ has measure zero. Removing this set from the lines we obtain a countable family of pairwise disjoint sets $\{\ell_i\setminus E\}_i$ with $\mu(\ell_i\setminus E)\geq\epsilon$. Hence $\infty>\mu(\mathbb{R}^2)\geq\mu(\bigcup_i (\ell_i\setminus E))=\infty$ which is a contradiction.