Aims: The kinematics, shaping, density distribution, expansion distance, and ionized mass of the nebula Hen 2-104, and the nature of its symbiotic Mira are investigated. Methods: A combination of multi-epoch HST images and VLT integral field high-resolution spectroscopy is used to study the nebular dynamics both along the line of sight and in the plane of the sky. These observations enable a 3-D spatio-kinematical model of the nebula to be constructed, which, with the measurement of its apparent expansion in the plane of the sky over a period of 4 years, provides the expansion parallax for the nebula. The integral-field data featuring the [Sii] λλ671.7, 673.1 emission line doublet provide a density map of the inner lobes of the nebula, which with the distance estimation provide a measurement of its ionized mass. Results: We measure densities ranging from n_e=500 to 1000 cm-3 in the inner lobes, and from 300 to 500 cm-3 in the outer lobes. We determine an expansion-parallax distance of 3.3 ± 0.9 kpc to Hen 2-104, which implies an unexpectedly large ionized mass for the nebula of approximately one tenth of a solar mass. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract No. NAS5-26555; and on observations obtained at the 4 m NTT and the 8 m VLT telescopes of the European Southern Observatory in Chile.