We analyse absorption characteristics and physical conditions of extraplanar intermediate- and high-velocity gas to study the distribution of the neutral and weakly ionised Milky Way halo gas and its relevance for the evolution of the Milky Way and other spiral galaxies. We combine optical absorption line measurements of Ca II/Na I and 21 cm emission line observations of HI along 103 extragalactic lines of sight towards quasars (QSOs) and active galactic nuclei (AGN). The archival optical spectra were obtained with the Ultraviolet and Visual Echelle Spectrograph (UVES) at the ESO Very Large Telescope, while the 21 cm HI observations were carried out using the 100-m radio telescope at Effelsberg. The analysis of the UVES spectra shows that single and multi-component Ca II/Na I absorbers at intermediate and high velocities are present in about 35 percent of the sight lines, indicating the presence of neutral extraplanar gas structures. In some cases the Ca II/Na I absorption is connected with HI 21 cm intermediate- or high-velocity gas with HI column densities in the range of 10 18 to 10 20 cm-2 (i.e., the classical IVCs and HVCs), while other Ca II/Na I absorbers show no associated HI emission. The observed HI line widths vary from ΔυFWHM=3.2 km s-1 to 32.0 km s-1 indicating a range of upper gas temperature limits of 250 K up to about 22 500 K. Our study suggests that the Milky Way halo is filled with a large number of neutral gaseous structures whose high column density tail represents the population of common HI high-velocity clouds seen in 21 cm surveys. The Ca II column density distribution follows a power-law f(N)=CNβ with a slope of β ≈ -1.6, thus comparable to the distribution found for intervening metal-line systems toward QSOs. Many of the statistical and physical properties of the Ca II absorbers resemble those of strong (W λ2796 > 0.3 Å) Mg II absorbing systems observed in the circumgalactic environment of other galaxies, suggesting that both absorber populations may be closely related.