The massive binary system Eta Carinae and the surrounding HII complex, the Carina Nebula, are potential particle acceleration sites from which very-high-energy (VHE; E > 100 GeV) gamma-ray emission could be expected. This paper presents data collected during VHE gamma-ray observations with the H.E.S.S. telescope array from 2004 to 2010, which cover a full orbit of Eta Carinae. In the 33.1-hour data set no hint of significant gamma-ray emission from Eta Carinae has been found and an upper limit on the gamma-ray flux of 7.7 x 10-13 ph cm-2 s-1 (99% confidence level) is derived above the energy threshold of 470 GeV. Together with the detection of high-energy (HE; 0.1 GeV > E > 100 GeV) gamma-ray emission by the Fermi-LAT up to 100 GeV, and assuming a continuation of the average HE spectral index into the VHE domain, these results imply a cut-off in the gamma-ray spectrum between the HE and VHE gamma-ray range. This could be caused either by a cut-off in the accelerated particle distribution or by severe gamma-gamma absorption losses in the wind collision region. Furthermore, the search for extended gamma-ray emission from the Carina Nebula resulted in an upper limit on the gamma-ray flux of 4.2 x 10-12 ph cm-2 s-1 (99% confidence level). The derived upper limit of ~23 on the cosmic-ray enhancement factor is compared with results found for the old-age mixed-morphology supernova remnant W 28.
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