Pulsars are precise clocks that can serve as Galactic accelerometers. Traditionally, the accelerations of stars within the Galaxy have been estimated using kinematic analyses that assume the Galaxy is in equilibrium. We use pulsar timing measurements to directly measure the Galactic acceleration. Given the measured Galactic acceleration, we derive the Oort limit for the first time without making any assumptions about spherical symmetry or equilibrium, and the local dark matter density given an accounting of the baryon budget. Our value for the local dark matter density is lower than typical Jeans estimates, and has implications for direct detection experiments. These measurements can also provide constraints on exotic forms of dark matter, such as collisional disk dark matter models. Finally, we obtain a constraint on the oblateness of the Milky Way potential, which implies that the pulsars trace the oblateness of the disk rather than the halo. Larger samples of pulsars should give us a direct view of dark matter sub-structure in the Milky Way. Extreme precision RV measurements towards pulsars may also constrain theories of gravity and post-Newtonian parameters.