Compact nodal soliton solutions for a Dirac equation with fractional nonlinearity are studied. This model is a natural field-theoretical generalization of the MIT bag model. In the limit where the power nonlinearity is arbitrarily small, the exact nodal solutions are obtained and shown to be different from the corresponding radial excitations in the MIT bag model. The complete energy spectrum is also different, except for massless confined fields in 1+1 dimensions. Finally the nonexistence of spherical solitons with negative parity is established.