In this paper, we present a formulation of the classical theory of fermionic (anticommuting) fields, which fits into the general framework proposed by Brunetti, Dütsch and Fredenhagen. It was inspired by the recent developments in perturbative algebraic quantum field theory and it also allows for a deeper structural understanding on the classical level. We propose a modification of this formalism that also allows to treat fermionic fields. In contrast to other formulations of classical theory of anticommuting variables, we do not introduce additional Grassman degrees of freedom. Instead the anticommutativity is introduced in a natural way on the level of functionals. Moreover, our construction incorporates the functional-analytic and topological aspects, which is usually neglected in the treatments of anticommuting fields. We also give an example of an interacting model where our framework can be applied.