Mathematical answer:
Let J be the total (vector) angular momentum operator and S[size=-4]1[/size], S[size=-4]2[/size] be the angular momentum operators for the first and second electron.
J² = J[size=-4]x[/size]² + J[size=-4]y[/size]² + J[size=-4]z[/size]²
= (S[size=-4]1x[/size] + S[size=-4]2x[/size])² + (S[size=-4]1y[/size] + S[size=-4]2y[/size])² + (S[size=-4]1z[/size] + S[size=-4]2z[/size])²
= S[size=-4]1[/size]² + S[size=-4]2[/size]² + 2 S[size=-4]1[/size] · S[size=-4]2[/size]
= S[size=-4]1[/size]² + S[size=-4]2[/size]² + 2 S[size=-4]1z[/size]S[size=-4]2z[/size] + S[size=-4]1+[/size]S[size=-4]2-[/size] + S[size=-4]1-[/size]S[size=-4]2+[/size]
(The last two terms are products of the raising and lowering operators, S[size=-4]±[/size] = S[size=-4]x[/size] ± iS[size=-4]y[/size])
Multiplying this expanded version of J² by the eigenket representing the state with two aligned spins (either |++> or |-->) gives
J² |++> = (S[size=-4]1[/size]² + S[size=-4]2[/size]² + 2 S[size=-4]1z[/size]S[size=-4]2z[/size] + S[size=-4]1+[/size]S[size=-4]2-[/size] + S[size=-4]1-[/size]S[size=-4]2+[/size]) |++>
= (3/4 + 3/4 + 2*1/2*1/2 + 0 + 0) hbar² |++>
= 2 hbar² |++>
So the |J| eigenvalue is sqrt(2) hbar.
Physical answer:
When we say the electron is a spin-1/2 particle, it (basically) means that the maximum component of the angular momentum S[size=-4]z[/size] in a given direction, as given by a measurement, is 1/2 hbar. The magnitude of the angular momentum |S| is actually sqrt(1/2*(1/2+1)) = sqrt(3)/2 hbar. This is a consequence of the Heisenberg Uncertainty Principle: none of the S[size=-4]x[/size], S[size=-4]y[/size], and S[size=-4]z[/size] operators commute with each other, so one cannot exactly measure all three components simultaneously. If the S[size=-4]z[/size] component were equal to |S|, you would know all three components exactly (the other two would have to be zero), violating the Uncertainty Principle. Anyway, this remains true when you add the angular momenta of two electrons together: the maximum value of J[size=-4]z[/size] is 1 hbar even though |J| is sqrt(2) hbar. If you're wondering why |J| is sqrt(2) hbar instead of sqrt(3) hbar, it is essentially the same reason: the angular momentum vectors of the two electrons cannot be in perfect alignment, as this would imply one could simultaneously measure J², S[size=-4]1z[/size], and S[size=-4]2z[/size] exactly, which is incompatible with the Uncertainty Principle.
