Why don't we "see" the classical Dirac field? The electromagnetic field describes photons. If there are many photons then things become classical and we can use classical electromagnetism to describe the EM field. We can also measure the EM field placing an antenna and watching the voltage in its terminals.
If the Dirac field describes electrons in a similar manner the EM field describes photons, why don't we see electronic waves or other classical field phenomena? What would have to be done to observe this phenomena?
 A: Fermions do not admit a "simple" classical approximation as boson fields or quantum mechanical (spinless non-relativistic) particles do, and this is due essentially to the exclusion principle (I cannot quantify this last assertion more without entering into details that are too complicated and technical for this context, however a classical limit fermionic field, even if mathematically definable, would essentially be intrinsically unobservable).
It is possible, however to collectively describe and observe classical effects for systems of many fermions. Mathematically, this is done performing a so-called multiscale semiclassical analysis: the idea is to consider a system with at the same time many fermions, and energy scales that are much bigger than Planck's scales. In other words, one considers simultaneously the limits $N\to \infty$, and $\hslash\to 0$ ($N$ being the number of fermions), in a suitable way (if I recall correctly, $\hslash\sim N^{-\frac{1}{3}}$ in three space dimensions). 
In this context, a classical plasma approximation is accurate in describing the system: the many electrons are effectively well described by a (classical) position and velocity distribution, obeying a transport equation called Vlasov equation.
