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: 2&pi;<sup>2</sup>/3.

For comparison: the volume of the Euclidean ball in R<sup>4</sup> is &pi;<sup>2</sup>/2 (which contradicts Mike's final statement).