We present Atacama large millimeter/submillimeter array (ALMA) and compact array (ACA) [CI]P-3(1) - P-3(0)([CI] (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 (H-2) outflow. We discover that the receding and approaching sides of the H-2 outflow, aligned east-west, exceed 10 kpc in their total extent. High resolution (0."24) [CI](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 [CI](1-0) and CO(1-0) data to constrain the CO-to-H-2 conversion factor (alpha CO) in the outflow, which is on average 2.1 +/- 1.2 M-circle dot (K km s(-1) pc(2))(-1). We estimate that 60 +/- 20% of the total H-2 gas reservoir of NGC 6240 is entrained in the outflow, for a resulting mass-loss rate of M-out = 2500 +/- 1200 M-circle dot 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 = 3.2 +/- 1.8 M-circle dot (K km s(-1) pc(2))(-1), which is at least twice the value commonly employed for (ultra) luminous infrared galaxies ((U) LIRGs). We observe a tentative trend of increasing r(21) LCO(2-1)'/LCO(1-0' ratios with velocity dispersion and measure r(21) > 1 in the outflow, whereas r(21) similar or equal to 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 alpha(CO) and increasing their r(21) values.