We introduce VLT-MUSE observations of the central 2'× 2' (30 × 30 pc) of the Tarantula Nebula in the Large Magellanic Cloud. The observations provide an unprecedented spectroscopic census of the massive stars and ionised gas in the vicinity of R136, the young, dense star cluster located in NGC 2070, at the heart of the richest star-forming region in the Local Group. Spectrophotometry and radial-velocity estimates of the nebular gas (superimposed on the stellar spectra) are provided for 2255 point sources extracted from the MUSE datacubes, and we present estimates of stellar radial velocities for 270 early-type stars (finding an average systemic velocity of 271 ± 41 km s-1). We present an extinction map constructed from the nebular Balmer lines, with electron densities and temperatures estimated from intensity ratios of the [S II], [N II], and [S III] lines. The interstellar medium, as traced by Hα and [N II] λ6583, provides new insights in regions where stars are probably forming. The gas kinematics are complex, but with a clear bi-modal, blue- and red-shifted distribution compared to the systemic velocity of the gas centred on R136. Interesting point-like sources are also seen in the eastern cavity, western shell, and around R136; these might be related to phenomena such as runaway stars, jets, formation of new stars, or the interaction of the gas with the population of Wolf-Rayet stars. Closer inspection of the core reveals red-shifted material surrounding the strongest X-ray sources, although we are unable to investigate the kinematics in detail as the stars are spatially unresolved in the MUSE data. Further papers in this series will discuss the detailed stellar content of NGC 2070 and its integrated stellar and nebular properties. Based on observations made with ESO telescopes at the Paranal observatory under programme ID 60.A-9351(A).Table 3 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/614/A147