Is it true that on the real line with the ordinary lebesgue measure, any set of positive measure is the union of intervals?

Regardless of what you mean by "intervals", I do not believe there is such a statement. If you restrict yourself to open intervals, the counterexample is simple:

the union of a set of positive measure between 1/4 and 3/4 with integers.

For closed intervals, a Cantor set of positive measure is a counterexample.

What you might have in mind is the statement of regularity:

A measurable set is a symetric difference of a Borel set and a set of measure 0. See

https://en.wikipedia.org/wiki/Regularity_theorem_for_Lebesgue_measure