21. Construct a new circumferential ring outside the given rectangle so that it has a known area of $\mathrm{m}^{2}$. For the problem to have a solution, what value should m take?

In the given text, the phrase "m个S取何值" is a bit ambiguous. A more precise translation might be "what value should m take for the area S?" However, I've kept it as close to the original as possible.
21. Let $\mathrm{a}, \mathrm{b}$ represent the sides of the known rectangle, and let $\phi$ represent the angle between the sides of the circumscribing rectangle and the known rectangle (Figure 23). In this case, the sides of the circumscribing rectangle will be:
$$
\mathrm{a} \cos \phi+\mathrm{b} \sin \phi \text { and } a \sin \phi+\mathrm{b} \cos \phi .
$$

According to the problem's conditions, we get:
$$
(a \cos \phi+b \sin \phi)(a \sin \phi+b \cos \phi)-\mathrm{m}^{2}
$$

From this, we can derive: $\sin 2 \phi=\frac{2\left(m^{2}-a b\right)}{a^{2}+b^{2}}$.
Therefore, the condition for the problem to be solvable is $0 \leqslant \sin 2 \phi \leqslant 1$, which is equivalent to the following inequality:
$$
\sqrt{a L} \leqslant \mathrm{~m} \leqslant \frac{a+b}{\sqrt{2}} .
$$