Modular forms of integral weight on G_2 and other exceptional groups were first studied by Gross-Wallach and Gan-Gross-Savin. These are special automorphic forms that appear to behave similarly to classical holomorphic modular forms; they have a semi-classical notion of Fourier expansion and Fourier coefficients. I will describe a theory of modular forms of half-integral weight on G_2 and other exceptional groups. Moreover, using the automorphic minimal representation on F_4 (as studied by Loke-Savin and Ginzburg), we define a modular form of weight 1/2 on G_2 whose Fourier coefficients are related to the 2-torsion in the narrow class groups of totally real cubic fields. This is joint work with Spencer Leslie.