Does every closed subset of the underlying space of a Noetherian scheme admits a natural closed subscheme structure?

For any scheme (you don't need noetherian) $X$ and a closed subset $i: Z \hookrightarrow X$ you can equip $Z$ with the structure of a reduced subscheme.

On an affine subset $\DeclareMathOperator{Spec}{Spec} U = \Spec A$ of $X$ you just take the sheaf $\mathcal{O}_{\Spec A/I_{U \cap Z}}$, where $I_{U \cap Z}$ is the radical ideal associated to $Z \cap U$. This makes $U \cap Z$ a closed subscheme of $U$.

You can then glue these sheafs together: If $U$ and $U' = \Spec A'$ are affine open subsets of $X$, cover their intersection by sets that are distinguished open in both $U$ and $U'$. Let $W = \Spec B$ be one of these. Then $\mathcal{O}_{\Spec A/I_{U \cap Z}}\vert_W$ and $\mathcal{O}_{\Spec A'/I_{U' \cap Z}}\vert_W$ both are isomorphic to $\mathcal{O}_{\Spec B/I_{W \cap Z}}$.