Using path-integral Monte Carlo simulations, we assess the core structure and mobility of the screw and edge basal-plane dislocations in hcp He4. Our findings provide key insights into recent interpretations of giant plasticity and mass flow junction experiments. First, both dislocations are dissociated into nonsuperfluid Shockley partial dislocations separated by ribbons of stacking fault, suggesting that they are unlikely to act as one-dimensional channels that may display Lüttinger-liquid-like behavior. Second, the centroid positions of the partial cores are found to fluctuate substantially, even in the absence of applied shear stresses. This implies that the lattice resistance to motion of the partial dislocations is negligible, consistent with the recent experimental observations of giant plasticity. Further results indicate that both the structure of the partial cores and the zero-point fluctuations play a role in this extreme mobility. © 2016 American Physical Society.1174