Aims. Our goal is to investigate the structure, elemental abundances, physical conditions, and the immediate surroundings of supernova remnant 0540-69.3 in the Large Magellanic Cloud. Methods. Imaging in [O III] and spectroscopic studies through various slits were carried out using European Souther Observatory’s Very Large and New Technology Telescopes. Densities, temperatures, and abundances were estimated applying nebular analysis for various parts of the remnant. Results. Several new spectral lines are identified, both from ejecta embedded in the pulsar-wind nebula, and in interstellar clouds shocked by the supernova blast wave. For the filaments in the pulsar-wind nebula, all lines are redshifted by 440 ± 80 km s−1 with respect to the rest frame of the host galaxy, and a 3D representation of the [O III] emission displays a symmetry axis of ring-like structures which could indicate that the pulsar shares the same general redshift as the central supernova ejecta. We note that [O II], [S II], [Ar III], and Hβ share a common more compact structure than [O III], and possibly [Ne III]. The average [O III] temperature for the filaments in the pulsar-wind nebula is 23 500 ± 1800 K, and the electron density derived from [S II] is typically ∼ 103 cm−3. By mass, the relative elemental abundances of the shocked ejecta in the pulsar-wind nebula are O : Ne : S : Ar ≈ 1 : 0.07 : 0.10 : 0.02, consistent with explosion models of 13 − 20 M⊙ progenitors, and similar to that of SN 1987A, as is also the explosive mixing of hydrogen and helium into the center. From Hβ and He Iλ5876, the mass ratio of He/H in the center is estimated to be in excess of ∼0.8. The rapid cooling of the shocked ejecta could potentially cause variations in the relative abundances if the ejecta are not fully microscopically mixed, and this is highlighted for S/O for the period 1989–2006. Also, [O III] is seen in presumably freely coasting photoionized ejecta outside the pulsar-wind nebula at inferred velocities out to well above 2000 km s−1, and in projection, [O III] is seen out to ∼10″ from the pulsar. This was used to estimate that the pulsar age is ≈1200 years. The freely coasting [O III]-emitting ejecta have a strictly nonspherical distribution, and their mass is estimated to be ∼0.12 M⊙. A possible outer boundary of oxygen-rich ejecta is seen in [O II] λλ3726,3729 at ∼2000 − 2100 km s−1. Four filaments of a shocked interstellar medium are identified, and there is a wide range in the degree of ionization of iron, from Fe+ to Fe13+. One filament belongs to a region also observed in X-rays, and another one has a redshift of 85 ± 30 km s−1 relative to the host. From this we estimate that the electron density of the [O III]-emitting gas is ∼ 103 cm−3, and that the line of the most highly ionized ion, [Fe XIV] λ5303, comes from an evaporation zone in connection with the radiatively cooled gas emitting, for example, [O III], and not from immediately behind the blast wave. We do not find evidence for nitrogen-enriched ejecta in the southwestern part of the remnant, as was previously suggested. Emission in this region is instead from a severely reddened H II-region.