The transmission and lateral shift properties of the one-dimensional photonic crystals containing Dirac semimetal-based hyperbolic metamaterials and chiral material defects are theoretically investigated by using the transfer matrix method in terahertz. It is found that the angle-independent omnidirectional photonic bandgaps (PBGs) for p-polarization can be realized in the photonic crystals made of dielectric and the Dirac semimetal-based hyperbolic metamaterials. However, the PBGs are blue-shifted with the incident angle for s-polarized wave. The omnidirectional PBGs can be tunable with the Fermi energy of the Dirac semimetal, the thickness ratio of the constitute layers and the structural parameters. When one or two isotropic chiral material layers are introduced in the photonic crystals, double or four polarization-dependent defect modes are found in the bandgap under oblique incidence due to the coupling and conversion between different polarized waves in the chiral defects. The frequency interval of the defect modes changes with the chirality parameter and the incident angle. It is also found that when a light is incident on the defective photonic crystals, the giant positive and negative Goos-Hänchen shifts of co-polarization reflected waves can be induced around the defect modes, which are sensitive to the polarization, the incident angle, the incident frequency and the chirality. The special optical properties of the proposed defective multiplayers provide possibilities for practical applications in designing polarization sensitive optical devices, such as filter and sensor.