51 Eridani b is an exoplanet around a young (20 Myr) nearby (29.4 pc) F0-type star, recently discovered by direct imaging. Being only 0.5" away from its host star it is well suited for spectroscopic analysis using integral field spectrographs. We aim to refine the atmospheric properties of this and to further constrain the architecture of the system by searching for additional companions. Using the SPHERE instrument at the VLT we extend the spectral coverage of the planet to the complete Y- to H-band range and provide photometry in the K12-bands (2.11, 2.25 micron). The object is compared to other cool and peculiar dwarfs. Furthermore, the posterior probability distributions of cloudy and clear atmospheric models are explored using MCMC. We verified our methods by determining atmospheric parameters for the two benchmark brown dwarfs Gl 570D and HD 3651B. For probing the innermost region for additional companions, archival VLT-NACO (L') SAM data is used. We present the first spectrophotometric measurements in the Y- and K-bands for the planet and revise its J-band flux to values 40% fainter than previous measurements. Cloudy models with uniform cloud coverage provide a good match to the data. We derive the temperature, radius, surface gravity, metallicity and cloud sedimentation parameter f_sed. We find that the atmosphere is highly super-solar (Fe/H~1.0) with an extended, thick cloud cover of small particles. The model radius and surface gravity suggest planetary masses of about 9 M_jup. The evolutionary model only provides a lower mass limit of >2 M_jup (for pure hot-start). The cold-start model cannot explain the planet's luminosity. The SPHERE and NACO/SAM detection limits probe the 51 Eri system at Solar System scales and exclude brown-dwarf companions more massive than 20 M_jup beyond separations of ~2.5 au and giant planets more massive than 2 M_jup beyond 9 au.