We introduce the 'Bluedisk' project, a large programme at the Westerbork Synthesis Radio Telescope that has mapped the H i in a sample of 23 nearby galaxies with unusually high H i mass fractions, along with a similar-sized sample of control galaxies. This paper presents the sample selection, observational set-up, data reduction strategy and a first analysis of the sizes and structural properties of the H i discs. We find that the H i-rich galaxies lie on the same H i mass versus H i size relation as normal spiral galaxies, extending it to total H i masses of 2 x 10(10) M-circle dot and radii R1 of similar to 100 kpc. The H i-rich galaxies have significantly larger values of H i-to-optical size ratio and more clumpy H i discs than those of normal spirals. There is no evidence that the discs of H i-rich galaxies are more disturbed. In fact, the centre of the H i distribution corresponds more closely with the centre of the optical light in the H i-rich galaxies than in the controls. All these results argue against a scenario in which new gas has been brought in by mergers. It is possible that they may be more consistent with cooling from a surrounding quasi-static halo of warm/hot gas.