Context. Understanding the formation mechanisms of protoplanetary disks and multiple systems and also their pristine properties are key questions for modern astrophysics. The properties of the youngest disks, embedded in rotating infalling protostellar envelopes, have largely remained unconstrained up to now. Aims. We aim to observe the youngest protostars with a spatial resolution that is high enough to resolve and characterize the progenitors of protoplanetary disks. This can only be achieved using submillimeter and millimeter interferometric facilities. In the framework of the IRAM Plateau de Bure Interferometer survey CALYPSO, we have obtained subarcsecond observations of the dust continuum emission at 231 GHz and 94 GHz for a sample of 16 solar-type Class 0 protostars. Methods. In an attempt to identify disk-like structures embedded at small scales in the protostellar envelopes, we modeled the dust continuum emission visibility profiles using Plummer-like envelope models and envelope models that include additional Gaussian disk-like components.Results. Our analysis shows that in the CALYPSO sample, 11 of the 16 Class 0 protostars are better reproduced by models including a disk-like dust continuum component contributing to the flux at small scales, but less than 25% of these candidate protostellar disks are resolved at radii > 60 au. Including all available literature constraints on Class 0 disks at subarcsecond scales, we show that our results are representative: most (> 72% in a sample of 26 protostars) Class 0 protostellar disks are small and emerge only at radii < 60 au. We find a multiplicity fraction of the CALYPSO protostars < ∼ 57% ± 10% at the scales 100-5000 au, which generally agrees with the multiplicity properties of Class I protostars at similar scales. Conclusions. We compare our observational constraints on the disk size distribution in Class 0 protostars to the typical disk properties from protostellar formation models. If Class 0 protostars contain similar rotational energy as is currently estimated for prestellar cores, then hydrodynamical models of protostellar collapse systematically predict a high occurrence of large disks. Our observations suggest that these are rarely observed, however. Because they reduce the centrifugal radius and produce a disk size distribution that peaks at radii < 100 au during the main accretion phase, magnetized models of rotating protostellar collapse are favored by our observations.