ABSTRACT Galactic bars are able to affect the evolution of galaxies by redistributing their gas, possibly contributing to the cessation of star formation. Several recent works point to ‘bar quenching’ playing an important role in massive disc galaxies. We construct a sample of six gas-rich and strongly barred disc galaxies with resolved H i observations. This sample of galaxies, which we call H i-rich barred galaxies, was identified with the help of Galaxy Zoo to find galaxies hosting a strong bar, and the Arecibo Legacy Fast Arecibo L-band Feed Array blind H i survey to identify galaxies with a high H i content. The combination of strong bar and high gas fraction is rare, so this set of six galaxies is the largest sample of its type with resolved H i observations. We measure the gas fractions, H i morphology and kinematics, and use archival optical data from the Sloan Digital Sky Survey to reveal star formation histories and bar properties. The galaxies with the lowest gas fractions (still very high for their mass) show clear H i holes, dynamically advanced bars, and low star formation rates, while those with the highest gas fractions show little impact from their bar on the H i morphology, and are still actively star-forming. These galaxies support a picture in which the movement of gas by bars can lead to star formation quenching. How these unusual galaxies came to be is an open question.