I found the answer in the paper "Measure and cardinality" by Briggs and Schaffter. In short: nonot if I interpret positive measure to mean positive outer measure. A proof is given that every measurable subset with cardinality less than that of $\mathbb{R}$ has Lebesgue measure zero. However, they then survey results of Solovay that show that there are models of ZFC in which CH fails and every subset of cardinality less than that of $\mathbb{R}$ is measurable, and that there are models of ZFC in which CH fails and there are subsets of cardinality $\aleph_1$ that are nonmeasurable. So it is undecidable in ZFC.
I found the answer in the paper "Measure and cardinality" by Briggs and Schaffter. In short: no. A proof is given that every measurable subset with cardinality less than that of $\mathbb{R}$ has Lebesgue measure zero. However, they then survey results of Solovay that show that there are models of ZFC in which CH fails and every subset of cardinality less than that of $\mathbb{R}$ is measurable, and that there are models of ZFC in which CH fails and there are subsets of cardinality $\aleph_1$ that are nonmeasurable. So it is undecidable in ZFC.