We present the highest resolution --- 15 pc (0.03) --- ALMA $^{12}$CO(2-1) line emission and 1.3mm continuum maps, tracers of the molecular gas and dust, respectively, in the nearby merging galaxy system NGC 6240, that hosts two supermassive black holes growing simultaneously. These observations provide an excellent spatial match to existing Hubble optical and near-infrared observations of this system. A significant molecular gas mass, $sim$9$times$10$^9$M$_odot$, is located in between the two nuclei, forming a clumpy stream kinematically dominated by turbulence, rather than a smooth rotating disk as previously assumed from lower resolution data. Evidence for rotation is seen in the gas surrounding the southern nucleus, but not in the northern one. Dynamical shells can be seen, likely associated with nuclear supernovae remnants. We further detect the presence of significant high velocity outflows, some of them reaching velocities $>$500 km/s, affecting a significant fraction, $sim$11% of the molecular gas in the nuclear region. Inside the spheres of influence of the northern and southern supermassive black holes we find molecular masses of 7.4$times$10$^8$M$_odot$ and 3.3$times$10$^9$M$_odot$, respectively. We are thus directly imaging the reservoir of gas that can accrete onto each supermassive black hole. These new ALMA maps highlight the critical need for high resolution observations of molecular gas in order to understand the feeding of supermassive black holes and its connection to galaxy evolution in the context of a major galaxy merger.
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