  The chemical yields of supernovae and the metal enrichment of the hot
intra-cluster medium (ICM) are not well understood. This paper introduces the
CHEmical Enrichment RGS Sample (CHEERS), which is a sample of 44 bright local
giant ellipticals, groups and clusters of galaxies observed with XMM-Newton.
This paper focuses on the abundance measurements of O and Fe using the
reflection grating spectrometer (RGS). The deep exposures and the size of the
sample allow us to quantify the intrinsic scatter and the systematic
uncertainties in the abundances using spectral modeling techniques. We report
the oxygen and iron abundances as measured with RGS in the core regions of all
objects in the sample. We do not find a significant trend of O/Fe as a function
of cluster temperature, but we do find an intrinsic scatter in the O and Fe
abundances from cluster to cluster. The level of systematic uncertainties in
the O/Fe ratio is estimated to be around 20-30%, while the systematic
uncertainties in the absolute O and Fe abundances can be as high as 50% in
extreme cases. We were able to identify and correct a systematic bias in the
oxygen abundance determination, which was due to an inaccuracy in the spectral
model. The lack of dependence of O/Fe on temperature suggests that the
enrichment of the ICM does not depend on cluster mass and that most of the
enrichment likely took place before the ICM was formed. We find that the
observed scatter in the O/Fe ratio is due to a combination of intrinsic scatter
in the source and systematic uncertainties in the spectral fitting, which we
are unable to disentangle. The astrophysical source of intrinsic scatter could
be due to differences in AGN activity and ongoing star formation in the BCG.
The systematic scatter is due to uncertainties in the spatial line broadening,
absorption column, multi-temperature structure and the thermal plasma models.
(Abbreviated).
