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 SF$_6$ at atmospheric pressure. Depending on the TPC readout technology, 10-20 helium recoils above 6 keVr or only 3-4 recoils above 20 keVr would suffice to distinguish a 10 GeV WIMP signal from the solar neutrino background. High-resolution 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 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.
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