  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.
