In other words, a (real) Banach space $X$ is Hilbert irreducible iff it has no $2$-dimensional subspace isometric to $\mathbb R^2$ with the Euclidean norm.
In $M_n(\mathbb R)$, the subspace $Y$ consisting of matrices whose entries below the first row are $0$ satisfies the parallelogram law.
The space $\mathbb c$ of real sequences converging to $0$ with supremum norm is Hilbert irreducible. To prove this, consider two linearly independent members $x$ and $y$ of $c$. It is easy to show that there is $\epsilon > 0$ such that $\|x + t y\|$ is an affine function of $t$ for $0 < t < \epsilon$. On the other hand, in $\mathbb R^2$ with Euclidean norm $\|(1,t)\|$ is strictly convex.