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We present Atacama large millimeter/submillimeter array (ALMA) and compact array (ACA) [C I]³P₁-³P₀ ([C I](1-0)) observations of NGC 6240, which we combine with ALMA CO(2-1) and IRAM Plateau de Bure Interferometer CO(1-0) data to study the physical properties of the massive molecular (H₂) outflow. We discover that the receding and approaching sides of the H₂ outflow, aligned east-west, exceed 10 kpc in their total extent. High resolution (0′′.24) [C I](1-0) line images surprisingly reveal that the outflow emission peaks between the two active galactic nuclei (AGNs), rather than on either of the two, and that it dominates the velocity field in this nuclear region. We combine the [C I](1-0) and CO(1-0) data to constrain the CO-to-H₂ conversion factor (α_CO) in the outflow, which is on average 2.1±1.2M_☉(Kkms^–1pc²)^–1. We estimate that 60 ± 20% of the total H₂ gas reservoir of NGC 6240 is entrained in the outflow, for a resulting mass-loss rate of {dot}M_out=2500±1200M_☉yr^–1≡50±30 SFR. These energetics rule out a solely star formation-driven wind, but the puzzling morphology challenges a classic radiative-mode AGN feedback scenario. For the quiescent gas, we compute <α_CO>=3.2±1.8M_☉(Kkms^–1pc²)^–1, which is at least twice the value commonly employed for (ultra) luminous infrared galaxies ((U)LIRGs). We observe a tentative trend of increasing r₂₁≡L_CO(2–1)^′^/LCO(1-0)_^′^ ratios with velocity dispersion and measure r₂₁ > 1 in the outflow, whereas r₂₁ ≃ 1 in the quiescent gas. We propose that molecular outflows are the location of the warmer, strongly unbound phase that partially reduces the opacity of the CO lines in (U)LIRGs, hence driving down their global α_CO and increasing their r₂₁ values.