Motivated by recent experimental realizations of polar metals with broken inversion symmetry, we explore the emergence of strong correlations driven by criticality when the polar transition temperature is tuned to zero. Overcoming previously discussed challenges, we demonstrate a robust mechanism for coupling between the critical mode and electrons in multiband metals. We identify and characterize several novel interacting phases, including non-Fermi liquids, when band crossings are close to the Fermi level and present their experimental signatures for three generic types of band crossings.