We report a comprehensive investigation of the magnetism of the $S$ = 3/2 triangular-lattice antiferromagnet, $alpha$-CrOOH(D) (delafossites green-grey powder). The nearly Heisenberg antiferromagnetic Hamiltonian ($J_1$ $sim$ 23.5 K) with a weak single-ion anisotropy of $|D|$/$J_1$ $sim$ 4.6% is quantitatively determined by fitting to the electron spin resonance (ESR) linewidth and susceptibility measured at high temperatures. The weak single-ion anisotropy interactions, possibly along with other perturbations, e.g. next-nearest-neighbor interactions, suppress the long-range magnetic order and render the system disordered, as evidenced by both the absence of any clear magnetic reflections in neutron diffraction and the presence of the dominant paramagnetic ESR signal down to 2 K ($sim$ 0.04$J_1$$S^2$), where the magnetic entropy is almost zero. The power-law behavior of specific heat ($C_m$ $sim$ $T^{2.2}$) observed below the freezing temperature of $T_f$ = 25 K in $alpha$-CrOOH or below $T_f$ = 22 K in $alpha$-CrOOD is insensitive to the external magnetic field, and thus is consistent with the theoretical prediction of a gapless U(1) Dirac quantum spin liquid (QSL) ground state. At low temperatures, the spectral weight of the low-energy continuous spin excitations accumulates at the K points of the Brillouin zone, e.g. $|mathbf{Q}|$ = 4$pi$/(3$a$), and the putative Dirac cones are clearly visible. Our work is a first step towards the understanding of the possible Dirac QSL ground state in this triangular-lattice magnet with $S$ = 3/2.
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