Context. Investigations of the magnetism of the quiet Sun are hindered by extremely weak polarization signals in Fraunhofer spectral lines. Photon noise, straylight, and the systematically different sensitivity of the Zeeman effect to longitudinal and transversal magnetic fields result in controversial results in terms of the strength and angular distribution of the magnetic field vector. Aims: The information content of Stokes measurements close to the diffraction limit of the 1.5 m GREGOR telescope is analyzed. We took the effects of spatial straylight and photon noise into account. Methods: Highly sensitive full Stokes measurements of a quiet-Sun region at disk center in the deep photospheric Fe i lines in the 1.56 μm region were obtained with the infrared spectropolarimeter GRIS at the GREGOR telescope. Noise statistics and Stokes V asymmetries were analyzed and compared to a similar data set of the Hinode spectropolarimeter (SOT/SP). Simple diagnostics based directly on the shape and strength of the profiles were applied to the GRIS data. We made use of the magnetic line ratio technique, which was tested against realistic magneto-hydrodynamic simulations (MURaM). Results: About 80% of the GRIS spectra of a very quiet solar region show polarimetric signals above a 3σ level. Area and amplitude asymmetries agree well with small-scale surface dynamo-magneto hydrodynamic simulations. The magnetic line ratio analysis reveals ubiquitous magnetic regions in the ten to hundred Gauss range with some concentrations of kilo-Gauss fields. Conclusions: The GRIS spectropolarimetric data at a spatial resolution of ≈0.̋4 are so far unique in the combination of high spatial resolution scans and high magnetic field sensitivity. Nevertheless, the unavoidable effect of spatial straylight and the resulting dilution of the weak Stokes profiles means that inversion techniques still bear a high risk of misinterpretating the data.