We present rest-frame optical emission-line flux ratio measurements for five z > 5 galaxies observed by the James Webb Space Telescope Near-Infared Spectrograph (NIRSpec) in the SMACS 0723 Early Release Observations. We add several quality-control and post-processing steps to the NIRSpec pipeline reduction products in order to ensure reliable relative flux calibration of emission lines that are closely separated in wavelength, despite the uncertain absolute spectrophotometry of the current version of the reductions. Compared to z similar to 3 galaxies in the literature, the z > 5 galaxies have similar [O iii]lambda 5008/H beta ratios, similar [O iii]lambda 4364/H gamma ratios, and higher (similar to 0.5 dex) [Ne III]lambda 3870/[O II]lambda 3728 ratios. We compare the observations to MAPPINGS V photoionization models and find that the measured [Ne III]lambda 3870/[O II]lambda 3728, [O iii]lambda 4364/H gamma, and [O iii]lambda 5008/H beta emission-line ratios are consistent with an interstellar medium (ISM) that has very high ionization ( log(Q)?8-9 , units of cm s(-1)), low metallicity (Z/Z (?) ? 0.2), and very high pressure ( log(P/k)?8-9, units of cm(-3)). The combination of [O iii]lambda 4364/H gamma and [O iii]lambda(4960 + 5008)/H beta line ratios indicate very high electron temperatures of 4.1 < log(T-e/K) < 4.4, further implying metallicities of Z/Z(?)?0.2 with the application of low-redshift calibrations for "T-e -based " metallicities. These observations represent a tantalizing new view of the physical conditions of the ISM in galaxies at cosmic dawn.