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We introduce a promising new platform for Majorana zero modes and various spintronics applications based on gate-defined wires in HgTe quantum wells. Because of the Dirac-like band structure for HgTe, the physics of such systems differs markedly from that of conventional quantum wires. Most strikingly, we show that the subband parameters for gate-defined HgTe wires exhibit exquisite tunability: Modest gate voltage variation allows one to modulate the Rashba spin-orbit energies from zero up to about 30 K, and the effective g factors from zero up to giant values exceeding 600. The large achievable spin-orbit coupling and g factors together allow one to access Majorana modes in this setting at exceptionally low magnetic fields while maintaining robustness against disorder. As an additional benefit, gate-defined wires (in HgTe or other settings) should greatly facilitate the fabrication of networks for refined transport experiments used to detect Majoranas, as well as the realization of non-Abelian statistics and quantum information devices.