I don't disagree that these sets are the same cardinality. But cardinality isn't the only way to describe the "size" of a set.

I suppose the typical measure-theoretic definition of "almost all" / "almost everywhere" insists on "everywhere but a zero-measure set", and you can't define a measure that satisfies sigma-additivity that treats intervals of finite Lebesgue measure as such, while ascribing nonzero measure to sets of infinite measure.

But even so, the Lebesgue measure of R is infinite, while the same measure of the unit interval is 1.