We describe a new approach to studying the intergalactic and circumgalactic medium in the local Universe: direct detection through narrow-band imaging of ultra-low surface brightness visible-wavelength line emission. We use the hydrodynamical cosmological simulation EAGLE to investigate the expected brightness of this emission at low redshift ($z$ $lesssim$ 0.2). H$alpha$ emission in extended halos (analogous to the extended Ly$alpha$ halos/blobs detected around galaxies at high redshifts) has a surface brightness of $gtrsim700$ photons cm$^{-2}$ sr$^{-1}$ s$^{-1}$ out to $sim$100 kpc. Mock observations show that the Dragonfly Telephoto Array, equipped with state-of-the-art narrow-band filters, could directly image these structures in exposure times of $sim$10 hours. H$alpha$ fluorescence emission from this gas can be used to place strong constraints on the local ultra-violet background, and on gas flows around galaxies. Detecting H$alpha$ emission from the diffuse intergalactic medium (the cosmic web) is beyond current capabilities, but would be possible with a hypothetical 1000-lens Dragonfly array.