Context. Dense molecular filaments are central to the star formation process, but the detailed manner in which they fragment into prestellar cores is not well understood yet. Aims. Here, we investigate the fragmentation properties and dynamical state of several star-forming filaments in the X-shaped nebula region of the California molecular cloud in an effort to shed some light on this issue. Methods. We used multiwavelength far-infrared images from Herschel as well as the getsources and getfilaments extraction methods to identify dense cores and filaments in the region and derive their basic properties. We also used a map of 13CO(2−1) emission from the Arizona 10m Submillimeter Telescope (SMT) to constrain the dynamical state of the filaments. Results. We identified ten filaments with aspect ratios of AR > 4 and column density contrasts of C > 0.5, as well as 57 dense cores, including two protostellar cores, 20 robust prestellar cores, 11 candidate prestellar cores, and 24 unbound starless cores. All ten filaments have roughly the same deconvolved full width at half maximum (FWHM), with a median value of 0.12 ± 0.03 pc, which is independent of their column densities ranging from <1021 cm−2 to >1022 cm−2. Two star-forming filaments (# 8 and # 10) stand out since they harbor quasi-periodic chains of dense cores with a typical projected core spacing of ~0.15 pc. These two filaments have thermally supercritical line masses and are not static. Filament 8 exhibits a prominent transverse velocity gradient, suggesting that it is accreting gas from the parent cloud gas reservoir at an estimated rate of ~40 ± 10 M⊙ Myr−1 pc−1. Filament 10 includes two embedded protostars with outflows and it is likely at a somewhat later evolutionary stage than filament 8. In both cases, the observed (projected) core spacing is similar to the filament width and significantly shorter than the canonical separation of ~4 times the filament width predicted by classical cylinder fragmentation theory. It is unlikely that projection effects can explain this discrepancy. We suggest that the continuous accretion of gas onto the two star-forming filaments, as well as the geometrical bending of the filaments, may account for the observed core spacing. Conclusions. Our findings suggest that the characteristic fragmentation lengthscale of molecular filaments is quite sensitive to external perturbations from the parent cloud, such as the gravitational accretion of ambient material.