Distributed acoustic sensing (DAS) is a rapidly developing technology particularly useful for the acquisition of vertical seismic profile (VSP) surveys. DAS data are increasingly used for seismic imaging, but not for estimating rock properties. We have developed a workflow for estimating elastic properties of the subsurface using full-waveform inversion (FWI) of DAS VSP data. Whereas conventional borehole geophones usually measure three components of particle velocity, DAS measures a single quantity, which is an approximation of the strain or strain rate along the fiber. Standard FWI algorithms are developed for particle velocity data, and hence their application to DAS data requires conversion of these data to particle velocity along the fiber. This conversion can be accomplished by a specially designed filter. Field measurements show that the conversion result is close to vertical particle velocity as measured by geophones. Elastic time-domain FWI of a synthetic multioffset VSP data set for a vertical well shows that the inversion of the vertical component alone is sufficient to recover elastic properties of the subsurface. Application of the proposed workflow to a multioffset DAS data set acquired at the CO2CRC Otway Project site in Victoria, Australia, reveals salient subhorizontal layering consistent with the known geology of the site. The inverted model at the well location matches the upscaled log with a correlation coefficient of 0.85.