Context. Cosmic rays and magnetic fields play an important role for the formation and dynamics of gaseous halos of galaxies. Aims: Low-frequency radio continuum observations of edge-on galaxies are ideal to study cosmic-ray electrons (CREs) in halos via radio synchrotron emission and to measure magnetic field strengths. Spectral information can be used to test models of CRE propagation. Free-free absorption by ionized gas at low frequencies allows us to investigate the properties of the warm ionized medium in the disk. Methods: We obtained new observations of the edge-on spiral galaxy NGC 891 at 129-163 MHz with the LOw Frequency ARray (LOFAR) and at 13-18 GHz with the Arcminute Microkelvin Imager (AMI) and combine them with recent high-resolution Very Large Array (VLA) observations at 1-2 GHz, enabling us to study the radio continuum emission over two orders of magnitude in frequency. Results: The spectrum of the integrated nonthermal flux density can be fitted by a power law with a spectral steepening towards higher frequencies or by a curved polynomial. Spectral flattening at low frequencies due to free-free absorption is detected in star-forming regions of the disk. The mean magnetic field strength in the halo is 7 ± 2 μG. The scale heights of the nonthermal halo emission at 146 MHz are larger than those at 1.5 GHz everywhere, with a mean ratio of 1.7 ± 0.3, indicating that spectral ageing of CREs is important and that diffusive propagation dominates. The halo scale heights at 146 MHz decrease with increasing magnetic field strengths which is a signature of dominating synchrotron losses of CREs. On the other hand, the spectral index between 146 MHz and 1.5 GHz linearly steepens from the disk to the halo, indicating that advection rather than diffusion is the dominating CRE transport process. This issue calls for refined modelling of CRE propagation. Conclusions: Free-free absorption is probably important at and below about 150 MHz in the disks of edge-on galaxies. To reliably separate the thermal and nonthermal emission components, to investigate spectral steepening due to CRE energy losses, and to measure magnetic field strengths in the disk and halo, wide frequency coverage and high spatial resolution are indispensable. LOFAR and AMI images (FITS files) are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/615/A98