With the increasing quantity and quality of data collected at volcanoes, there is growing potential to incorporate all the data into analyses of the magmatic system. Physics-based models provide a natural and meaningful way to bring together real-time monitoring data and laboratory analyses of eruption products, at the same time improving our understanding of volcanic processes. We develop a framework for joint inversions of diverse time series data using the physics-based model for dome-forming eruptions from \citeA{Wong2019}. Applying this method to the 2004-2008 eruption at Mount St. Helens, we estimate essential system parameters including chamber geometry, pressure, volatile content and material properties, from extruded volume, ground deformation and carbon dioxide emissions time series. The model parameter space is first sampled using the neighborhood search algorithm, then the resulting ensemble of models is resampled to generate posterior probability density functions on the parameters \cite{Sambridge1999_Search, Sambridge1999_Appraise}. We find models that fit all three datasets well. Posterior PDFs suggest an elongate chamber with aspect ratio less than 0.55, located at $9.0-17.2$ km depth. Since the model calculates pressure change during the eruption, we can constrain chamber volume to $64-256$ km$^3$. Volume loss in the chamber is $20-66$ million m$^3$. At the top of the chamber, total (dissolved and exsolved) water contents are $4.99-6.44$ wt\% and total carbon dioxide contents are $1560-3891$ ppm, giving a porosity of 5.3-16.6\% depending on the conduit length. Compared to previous inversions using a steady-state conduit model, we obtain a lower magma permeability scale, radius and friction coefficient.