Why are Grassmann fields never classical?

Well, it depends on what is meant by the word classical.

1. Usually in physics, classical theories mean theories where Planck's constant $\hbar$ is zero. If that's what is meant, then there certainly exist classical Grassmann-odd variables/fermions and a (in that sense) classical notion of supermanifolds. See e.g. this Phys.SE post.

2. What the authors may refer to is the weird nature of Grassmann-odd variables. Examples:

   • It is not possible to physically measure a Grassmann-odd variable in some detector.

   • There is no non-trivial topology in Grassmann-odd directions.

   • It is not possible to define a definite Berezin integral on a proper subset.

   • See also e.g. this Phys.SE post.

Let me put things in broader perspective. Bosonic fields are quantized in terms of commutators with a prefactor $\hbar$. Classical limit leads to commuting variables that may be represented by complex numbers. This is by the way the first step to devise numerical applications of path integrals. Fermionic fields require anticommutators ( consider 4d to focus and avoid special features of low dimension ) and formal $\hbar\to 0$ gives Grassmann algebra, say, $\{\theta_i, \theta_j\}=0$ and these variables cannot be represented as complex numbers ( just take $i=j$) in any simple way.