The electric and magnetic properties of matter are of great interest for materials science and their use in electronic applications. Large dielectric and magnetoelectric responses of materials at room temperature are a great advantage for electromagnetic device applications. Here we present a study of FeCo-MgF nanogranular films exhibiting giant dielectric and magnetoelectric responses at room temperature; with dielectric constant ε′ = 490 and magnetoelectric response Δε′/ε′0 = 3%. In these films, Fe-Co alloy-based nanometer-sized magnetic granules are dispersed in a Mg-fluoride-based insulator matrix. Insulating nanogranular films are a new class of multifunctional materials. The giant responses are caused by spin-dependent charge oscillation between magnetic granules via quantum-mechanical tunnelling. A possible application of such insulating nanogranular materials with giant response is in the construction of a tunable device, in which impedance components such as capacitance and inductance are tunable at room temperature.