Concerning the 2x2 case: As Mike points out, you can write down an explicit formula for the norm of the matrix {{a,b},{c,d}}. It takes a good while but Mathematica can then compute the volume you're asking for.

Integrate[If[a^2 + b^2 + c^2 + d^2
 + Sqrt[((b+c)^2 + (a-d)^2) ((b-c)^2 + (a+d)^2)] <= 2, 1, 0],
{a, -1, 1}, {b, -1, 1}, {c, -1, 1}, {d, -1, 1}]

It's answer is: 2π²/3.

For comparison: the volume of the Euclidean ball in R⁴ is π²/2 (which contradicts Mike's final statement).

Concerning the 2x2 case: As Mike points out, you can write down an explicit formula for the norm of the matrix {{a,b},{c,d}}. It takes a good while but Mathematica can then compute the volume you're asking for.

Integrate[If[a^2 + b^2 + c^2 + d^2
 + Sqrt[((b+c)^2 + (a-d)^2) ((b-c)^2 + (a+d)^2)] <= 2, 1, 0],
{a, -1, 1}, {b, -1, 1}, {c, -1, 1}, {d, -1, 1}]

Its answer is: 2π²/3.

For comparison: the volume of the Euclidean ball in R⁴ is π²/2 (which contradicts Mike's final statement that the matrix norm ball sits inside the Euclidean one).