Now that conventional weakly interacting massive particle (WIMP) dark matter searches are approaching the neutrino floor, there has been a resurgence of interest in detectors with sensitivity to nuclear recoil directions. A large-scale directional detector is attractive in that it would have sensitivity below the neutrino floor, be capable of unambiguously establishing the galactic origin of a purported dark matter signal, and could serve a dual purpose as a neutrino observatory. We present the first detailed analysis of a 1000 m 3 -scale detector capable of measuring a directional nuclear recoil signal at low energies. We propose a modular and multi-site observatory consisting of time projection chambers (TPCs) filled with helium and SF6 at atmospheric pressure. By comparing several available readout technologies, we identify high-resolution strip readout TPCs as the optimal tradeoff between performance and cost. We estimate that suitable angular resolution and head-tail recognition is achievable down to helium recoil energies of ∼6 keVr. Depending on the readout technology, an average of only 4-5 detected 100-GeV c −2 WIMP-fluorine recoils above 50 keVr are sufficient to rule out an isotropic recoil distribution at 90% CL. An average of 10-20 helium recoils above 6 keVr or only 3-4 helium recoils above 20 keVr would suffice to distinguish a 10 GeV c −2 WIMP signal from the solar neutrino background. High-resolution TPC charge readout also enables powerful electron background rejection capabilities well below 10 keV. We detail background and site requirements at the 1000 m 3 -scale, and identify materials that require improved radiopurity. The final experiment, which we name Cygnus-1000, will be able to observe ∼ 10-40 neutrinos from the Sun, depending on the final energy threshold. With the same exposure, the sensitivity to spin independent cross sections will extend into presently unexplored sub-10 GeV c −2 parameter space. For spin dependent interactions, already a 10 m 3 -scale experiment could compete with upcoming generation-two detectors, but Cygnus-1000 would improve upon this considerably. Larger volumes would bring sensitivity to neutrinos from an even wider range of sources, including galactic supernovae, nuclear reactors, and geological processes.