Is the cross product of two unit vectors itself a unit vector?

Or, in general, what does the magnitude of the cross product mean? How would you prove or disprove this?

No - for example, the cross product of any unit vector with itself is 0. In general, the magnitude of the cross product of vectors $\vec{a}$ and $\vec{b}$ is $$|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\sin(\theta)$$ where $\theta$ is the angle between the vectors $\vec{a}$ and $\vec{b}$. Thus, the cross product of two unit vectors $\vec{u}$ and $\vec{v}$ is itself a unit vector if and only if $\vec{u}$ and $\vec{v}$ are orthogonal, i.e. meet at right angles (this makes $\sin(\theta)=\sin(\frac{\pi}{2})=1$).

As to the general interpretation of the magnitude of the cross product, see Wikipedia:

The magnitude of the cross product can be interpreted as the positive area of the parallelogram having $a$ and $b$ as sides.

$|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}||\sin(\theta)|$

Let $a,b$ unit vectors, so we have $|a| = |b| = 1$

$|\vec{a}\times\vec{b}|=|\sin(\theta)| \le 1$ (equality is when $|\sin(\theta)| = 1$ i.e. when a and b are perpendicular)

Therefore in general the result won't be a unit vector.