Does a set with strictly positive Lebesgue measure contain an interval?

Nope. There is a standard construction of "fat Cantor sets" which have positive measure but are in fact nowhere dense, which is considerably stronger than merely containing no intervals. The construction proceeds much like constructing a Cantor set with zero measure except that the lengths of the deleted intervals decay more rapidly, and so they sum up to less than the total measure of the interval you started with. See https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set for details.

The irrationals are an easy example of not having this property.