  The GeMSE (Germanium Material and meteorite Screening Experiment) facility
operates a low-background HPGe crystal in an underground laboratory with a
moderate rock overburden of 620$\,$m.w.e. in Switzerland. It has been optimized
for continuous remote operation. A multi-layer passive shielding, a muon veto,
and a boil-off nitrogen purge line inside the measurement cavity minimize the
instrument's background rate, which decreased by 33% to
(164$\pm$2)$\,$counts/day (100-2700$\,$keV) after five years of underground
operation. This agrees with the prediction based on the expected decay of
short-lived isotopes. A fit to the known background components, modeled via a
precise simulation of the detector, shows that the GeMSE background is now
muon-dominated. We also present updates towards a more accurate detection
efficiency calculation for the screened samples: the thickness of the crystal's
outer dead-layer is precisely determined and the efficiency can now be easily
calculated for any sample geometry. The advantage of this feature is showcased
via the determination of the $^{40}$K content in the screening of a
complex-shaped object: a banana.
