We present a study of molecular-hydrogen absorption lines in the continuum spectra of three early-type stars in the Carina Nebula, HD 93129a, HD 93250, and HD 303308, made with the Berkeley Extreme and Far-Ultraviolet Spectrometer (BEFS) on the ORFEUS telescope in 1993 September. The instrument possesses an intermediate spectral resolving power (lambda/Delta lambda = 3000) over the wavelength range lambda lambda 390-1170. We derive column densities for the H-2 rotational levels up to J" = 6 and estimate the kinetic temperature and the UV radiation fields at the surface of the clouds in the nebula. All three stars show strong H-2 absorption features, indicating that the Carina Nebula contains abundant molecular hydrogen. The UV radiation field strength seems to be correlated with the angular distance of the target star from eta Car, implying that the clouds might be located close to eta Car. We derive the absolute distances between the clouds and eta Car, which are in good agreement with the angular positions of these stars. We also examine the morphology of the Carina Nebula based on the present result, as well as previous CO observations, and derive the conversion factor N(H-2)/W((CO)-C-12) in this region.