Context. Low-frequency radio continuum observations (<300 MHz) can provide valuable information on the propagation of low-energy cosmic ray electrons (CRE). Nearby spiral galaxies have hardly been studied in this frequency range because of the technical challenges of low-frequency radio interferometry. This is now changing with the start of operations of LOFAR. Aims. We aim to study the propagation of low-energy CRE in the interarm regions and the extended disk of the nearly face-on spiral galaxy Messier 51. We also search for polarisation in M 51 and other extragalactic sources in the field. Methods. The grand-design spiral galaxy M 51 was observed with the LOFAR High Frequency Antennas (HBA) and imaged in total intensity and polarisation. This observation covered the frequencies between 115 MHz and 175 MHz with 244 subbands of 8 channels each, resulting in 1952 channels. This allowed us to use RM synthesis to search for polarisation. Results. We produced an image of total emission of M 51 at the mean frequency of 151 MHz with 20′′ resolution and 0.3 mJy rms noise, which is the most sensitive image of a galaxy at frequencies below 300 MHz so far. The integrated spectrum of total radio emission is described well by a power law, while flat spectral indices in the central region indicate thermal absorption. We observe that the disk extends out to 16 kpc and see a break in the radial profile near the optical radius of the disk. The radial scale lengths in the inner and outer disks are greater at 151 MHz, and the break is smoother at 151 MHz than those observed at 1.4 GHz. The arm-interarm contrast is lower at 151 MHz than at 1400 MHz, indicating propagation of CRE from spiral arms into interarm regions. The correlations between the images of radio emission at 151 MHz and 1400 MHz and the FIR emission at 70 μm reveal breaks on scales of 1.4 and 0.7 kpc, respectively. The total (equipartition) magnetic field strength decreases from about 28 μG in the central region to about 10 μG at 10 kpc radius. No significant polarisation was detected from M 51, owing to severe Faraday depolarisation. Six extragalactic sources are detected in polarisation in the M 51 field of 4.1◦ × 4.1◦ size. Two sources show complex structures in Faraday space. Conclusions. Our main results, the scale lengths of the inner and outer disks at 151 MHz and 1.4 GHz, arm-interarm contrast, and the break scales of the radio-FIR correlations, can be explained consistently by CRE diffusion, leading to a longer propagation length of CRE of lower energy. The distribution of CRE sources drops sharply at about 10 kpc radius, where the star formation rate also decreases sharply. We find evidence that thermal absorption is primarily caused by H ii regions. The non-detection of polarisation from M 51 at 151 MHz is consistent with the estimates of Faraday depolarisation. Future searches for polarised emission in this frequency range should concentrate on regions with low star formation rates.