No subset of ℝ with a finite length can be expressed as a disjoint union of uncountably many sets where each set has a positive length. But I don't know if the same holds for ℝ.
I have tried this approach
Assuming that ℝ has an expression of this form we can show that there exists an uncountable subcover of the considered cover of ℝ such that it covers [n,n+1] for some n ∈ℕ, and it is the smallest subcover(from the assumed cover of ℝ) of [n,n+1]. If it is possible to prove that there exists an uncountable subcover which equals [n,n+1] I will be done, but I am not able to prove that there exists an uncountable subcover which is equal to [n,n+1].