Context. Barium stars are peculiar red giants characterized by an overabundance of the elements synthesized in the slow neutron-capture nucleosynthesis (s-process elements) along with an enrichment in carbon. These stars are discovered in binaries with white dwarf companions. The more recently formed of these stars are still surrounded by a planetary nebula. Aims. Precise abundance determinations of the various s-process elements, of further key elements that act as indicators for effectiveness of nucleosynthesis on the asymptotic giant branch and, especially, of the lightest, short-lived radionuclide technetium will establish constraints for the formation of s-process elements in asymptotic giant branch stars as well as mass transfer through, for example, stellar wind, Roche-lobe overflow, and common-envelope evolution. Methods. We performed a detailed spectral analysis of the K-type subgiant central star of the planetary nebula Hen 2−39 based on high-resolution optical spectra obtained with the Ultraviolet and Visual Echelle Spectrograph at the Very Large Telescope using local thermodynamic equilibrium model atmospheres. Results. We confirm the effective temperature of T_eff = (4350 ± 150) K for the central star of the planetary nebula Hen 2−39. It has a photospheric carbon enrichment of [C∕H] = 0.36 ± 0.08 and a barium overabundance of [Ba∕Fe] = 1.8 ± 0.5. We find a deficiency for most of the iron-group elements (calcium to iron) and establish an upper abundance limit for technetium (log ɛ_Tc < 2.5). Conclusions. The quality of the available optical spectra is not sufficient to measure abundances of all s-process elements accurately. Despite large uncertainties on the abundances as well as on the model yields, the derived abundances are most consistent with a progenitor mass in the range 1.75–3.00 M_⊙ and a metallicity of [Fe∕H] = −0.3 ± 1.0. This result leads to the conclusion that the formation of such systems requires a relatively large mass transfer that is most easily obtained via wind-Roche lobe overflow.