The volume of such a pyramid is zero; the four points are coplanar. To see why, let's work with coordinates. Let A be at (0,0,0), B at (p,0,0), C at (0,q,0), and D at (x,y,z).
We have AD=r, so x²+y²+z²=r²; BD=q, so (x-p)²+y²+z²=q²; and CD=p, so x²+(y-q)²+z²=p².
Subtracting BD from AD, 2px-x²=p², so x=p. Subtracting CD from AD, 2qy-q²=q², so y=q. Finally, substituting these values in AD, p²+q²+z²=r², so z=0, and D lies in the same plane as A, B and C.