NH and OH stretching excitations of hydrated double-stranded DNA oligomers containing guanine-cytosine (GC) base pairs in a Watson-Crick geometry are studied by two-dimensional (2D) infrared spectroscopy. The 2D spectra measured at a low hydration level (similar to 4 water molecules/base pair) are dominated by NH stretch contributions from the NH2 groups of G and C and the NH group of G. Partially hydrated NH2 groups display red-shifted NH stretch frequencies and a mixing of the wave functions of the two local NH oscillators via the mechanical vibrational coupling. The NH stretch lifetimes are of the order of 200-300 fs. Weak couplings exist between NH stretch oscillators within a base pair, while interactions between neighboring GC pairs in the double helix are negligible. The absence of spectral diffusion on a 1 ps time scale suggests a relatively rigid structure of the hydrogen bonds between DNA and residual water molecules. 2D spectra recorded with fully hydrated DNA oligomers exhibit NH and OH stretch contributions with a weak influence of water fluctuations on the NH stretch lineshapes. The femtosecond spectral diffusion of OH stretch excitations is slower than that in bulk H2O and originates from structural fluctuations of the water shell and the formation of a vibrationally hot ground state by vibrational relaxation. We compare our findings with measurements on hydrated adenine-thymine DNA oligomers and anhydrous GC base pairs in solution.