Without an effective prophylactic solution, infections from SARS-CoV-2 continue to rise worldwide with devastating health and economic costs. SARS-CoV-2 gains entry into host cells via an interaction between its Spike protein and the host cell receptor angiotensin converting enzyme 2 (ACE2). Disruption of this interaction confers potent neutralization of viral entry, providing an avenue for vaccine design and for therapeutic antibodies. Here, we develop single-domain antibodies (nanobodies) that potently disrupt the interaction between the SARS-CoV-2 Spike and ACE2. By screening a yeast surface-displayed library of synthetic nanobody sequences, we identified a panel of nanobodies that bind to multiple epitopes on Spike and block ACE2 interaction via two distinct mechanisms. Cryogenic electron microscopy (cryo-EM) revealed that one exceptionally stable nanobody, Nb6, binds Spike in a fully inactive conformation with its receptor binding domains (RBDs) locked into their inaccessible down-state, incapable of binding ACE2. Affinity maturation and structure-guided design of multivalency yielded a trivalent nanobody, mNb6-tri, with femtomolar affinity for SARS-CoV-2 Spike and picomolar neutralization of SARS-CoV-2 infection. mNb6-tri retains stability and function after aerosolization, lyophilization, and heat treatment. These properties may enable aerosol-mediated delivery of this potent neutralizer directly to the airway epithelia, promising to yield a widely deployable, patient-friendly prophylactic and/or early infection therapeutic agent to stem the worst pandemic in a century.