The spin of an electron is not meant in the literal sense (ie like some sort of spinning top or rotating sphere). Instead, it is a vector quantity and we define a spin quantum number.

I believe a sort of rough calculation goes as follows:
[tex] \tau = \frac{dL}{dt} = Iα [/tex]
If we define [itex] L = m_l\hbar = \frac{\hbar}{2}[/itex] and [itex] I = \frac{2}{5}mr^2, [/itex] an electron as a solid sphere with [itex] r ≈ fm[/itex]

Then, [tex] L = Iω = \frac{Iv}{r} => v = \frac{Lr}{I} = \frac{5Lr}{2mr^2}. [/tex]
Simplifying and inputting known data gives v ≈ 10^{11} ms^{-1}, which is 1000 times the speed of light.

The above calculation assumes that the electron is behaving like a rotating sphere and so it is meant to demonstrate why we should not think of spin as a classical concept here.

The electron is observed to be pointlike, which means its size, if it has one, is smaller than we have so far been able to detect. (10^{-16} cm). As a particle, it has a number of properties: mass, charge and angular momentum. Angular momentum comes in two varieties: orbital angular momentum (r x p) and spin angular momentum. There is no motion associated with spin angular momentum, and so the electron does not rotate. The angular momentum it carries (ħ/2) is simply an intrinsic property.