It is known that open sets in real line can be written as a countable union of disjoint open intervals. (link) I'm curious that if there is similar statements for closed sets in real line.
Closed sets are the inverse of open sets. So the correct way to write it would be the intersection of closed sets; however intersecting disjoint sets is boring. So the correct way to think about it would be that every closed set can be written as an intersection of closed sets whose complements are pairwise disjoint.
If you are indeed interested in unions then note that every closed set is the union of itself, trivially. If you want to ask about the union of closed intervals, then the answer is no. Since some closed sets (e.g. the Cantor set) does not contain any interval. Unless you consider singletons as closed intervals, in which case you would need uncountably many and the question becomes somewhat moot (every set is the union of singletons).
Moreover Sierpinski proved that if $\Bbb R$ (or any Baire space) is the countable union of disjoint closed sets then exactly one is non-empty. So you cannot get a nontrivial result using union of disjoint closed sets.