Supermassive black hole binary systems form in galaxy mergers and reside in galactic nuclei with large and poorly constrained concentrations of gas and stars. These systems emit nanohertz gravitational waves that will be detectable by pulsar timing arrays. Here we estimate the properties of the local nanohertz gravitational-wave landscape that includes individual supermassive black hole binaries emitting continuous gravitational waves and the gravitational-wave background that they generate. Using the 2 Micron All-Sky Survey, together with galaxy merger rates from the Illustris simulation project, we find that there are on average 91 ± 7 continuous nanohertz gravitational-wave sources, and 7 ± 2 binaries that will never merge, within 225 Mpc. These local unresolved gravitational-wave sources can generate a departure from an isotropic gravitational-wave background at a level of about 20 per cent, and if the cosmic gravitational-wave background can be successfully isolated, gravitational waves from at least one local supermassive black hole binary could be detected in 10 years with pulsar timing arrays.