We present high-resolution observations of the atomic and molecular gas phase of the "superwind" galaxy NGC 4666. Deep Very Large Array (VLA) H I observations of NGC 4666 and its surroundings reveal the presence of prominent tidal arms, which provide clear evidence that the galaxy is interacting with its neighbor NGC 4668 and a newly discovered dwarf companion. This interaction is also evident from a dynamical analysis, which shows that the diffuse H I envelope around NGC 4666 (radii >19 kpc) is kinematically altered with regard to the central disk. This interaction has likely caused the starburst activity in NGC 4666. A cut through the H I emission distribution perpendicular to the major axis provides no evidence for the existence of an H I halo: this distribution can best be described by an inner disk and two outer spiral arms. Our high-resolution Owens Valley Radio Observatory (OVRO) CO mosaic shows that the molecular gas is distributed relatively uniformly in the central ~7 kpc; the total molecular gas mass is ~1.0 × 10¹⁰ M. A kinematically interesting CO feature that can be interpreted as an expanding molecular supershell is detected near the footpoint of one of the Hα outflows at the turnover of galactic rotation. About 1% of the total energy input of the starburst would be needed to create this feature. A study of the CO (2 → 1) to CO (1 → 0) line-transition ratio (obtained with the IRAM 30 m and Swedish-ESO Submillimeter Telescope [SEST] telescopes) shows that there is no significant variation of the molecular gas excitation over the central galaxy disk [S(2 → 1)/S(1 → 0) 0.85] on scales of ~2.5 kpc. This can be explained by the unusually uniform star formation rate across NGC 4666's disk, which sets this system apart from most other starburst systems. The total amount of molecular and atomic gas is M_mol 3.0 × 10¹⁰ M, which implies that NGC 4666 can sustain many similar starburst episodes (consuming ~10⁸ M each) in the future. Two new dwarf companions in the NGC 4666 group are detected, with a detection limit of 10⁷ M, over an area of ~1.6 × 10⁴ kpc². We speculate that the huge H I envelope around NGC 4666 may represent a low-redshift counterpart for damped Lyα systems seen at higher redshifts.