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 ~ 3 galaxies in the literature, the z > 5 galaxies have similar [O III]λ5008/Hβ ratios, similar [O III]λ4364/Hγ ratios, and higher (~0.5 dex) [Ne III]λ3870/[O II]λ3728 ratios. We compare the observations to MAPPINGS V photoionization models and find that the measured [Ne III]λ3870/[O II]λ3728, [O III]λ4364/Hγ, and [O III]λ5008/Hβ emission-line ratios are consistent with an interstellar medium (ISM) that has very high ionization ( $\mathrm{log}(Q)\simeq 8-9$ , units of cm s-1), low metallicity (Z/Z ⊙ ≲ 0.2), and very high pressure ( $\mathrm{log}(P/k)\simeq 8-9$ , units of cm-3). The combination of [O III]λ4364/Hγ and [O III]λ(4960 + 5008)/Hβ line ratios indicate very high electron temperatures of $4.1\lt \mathrm{log}({T}_{e}/{\rm{K}})\lt 4.4$ , 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.