High-energy emission associated with star formation has been proposed as a significant source of interstellar medium (ISM) ionization in low-metallicity starbursts and an important contributor to the heating of the intergalactic medium (IGM) in the high-redshift (z>~8) universe. Using Chandra observations of a sample of 30 galaxies at D~200-450Mpc that have high specific star formation rates of 3-9Gyr^-1^ and metallicities near Z~0.3Z_{sun}_, we provide new measurements of the average 0.5-8keV spectral shape and normalization per unit star formation rate (SFR). We model the sample-combined X-ray spectrum as a combination of hot gas and high-mass X-ray binary (HMXB) populations and constrain their relative contributions. We derive scaling relations of logL_0.5-8keV_^HMXB^/SFR=40.19+/-0.06 and logL_0.5-2keV_^gas^/SFR=39.58_-0.28_^+0.17^; significantly elevated compared to local relations. The HMXB scaling is also somewhat higher than L_0.5-8keV_^HMXB^-SFR-Z relations presented in the literature, potentially due to our galaxies having relatively low HMXB obscuration and young and X-ray luminous stellar populations. The elevation of the hot gas scaling relation is at the level expected for diminished attenuation due to a reduction of metals; however, we cannot conclude that an L_0.5-2keV_^gas^-SFR-Z relation is driven solely by changes in ISM metal content. Finally, we present SFR-scaled spectral models (both emergent and intrinsic) that span the X-ray-to-IR band, providing new benchmarks for studies of the impact of ISM ionization and IGM heating in the early universe.