If a subset of the real line is uncountable and closed, does it have to contain a closed interval? Is there any theorem related to this?
No, this is not the case.
The Cantor set is the classic example. But it is true that if a subset of $\Bbb R$ is closed and uncountable, then it contains a copy of the Cantor set, in particular every closed uncountable subset of $\Bbb R$ has the same cardinality as $\Bbb R$ itself.