Context. Extremely metal-poor (XMP) galaxies are chemically, and possibly dynamically, primordial objects in the local Universe. Aims: Our objective is to characterize the H i content of the XMP galaxies as a class, using as a reference the list of 140 known local XMPs compiled by Morales-Luis et al. (2011). Methods: We have observed 29 XMPs, which had not been observed before at 21 cm, using the Effelsberg radio telescope. This information was complemented with H i data published in literature for a further 53 XMPs. In addition, optical data from the literature provided morphologies, stellar masses, star-formation rates and metallicities. Results: Effelsberg H i integrated flux densities are between 1 and 15 Jy km s-1, while line widths are between 20 and 120 km s-1. H i integrated flux densities and line widths from literature are in the range 0.1-200 Jy km s-1 and 15-150 km s-1, respectively. Of the 10 new Effelsberg detections, two sources show an asymmetric double-horn profile, while the remaining sources show either asymmetric (seven sources) or symmetric (one source) single-peak 21 cm line profiles. An asymmetry in the H i line profile is systematically accompanied by an asymmetry in the optical morphology. Typically, the g-band stellar mass-to-light ratios are ~0.1, whereas the H i gas mass-to-light ratios may be up to two orders of magnitude larger. Moreover, H i gas-to-stellar mass ratios fall typically between 10 and 20, denoting that XMPs are extremely gas-rich. We find an anti-correlation between the H i gas mass-to-light ratio and the luminosity, whereby fainter XMPs are more gas-rich than brighter XMPs, suggesting that brighter sources have converted a larger fraction of their H i gas into stars. The dynamical masses inferred from the H i line widths imply that the stellar mass does not exceed 5% of the dynamical mass, while the H i mass constitutes between 20 and 60% of the dynamical mass. Furthermore, the dark matter mass fraction spans a wide range, but can account, in some cases, for over 65% of the dynamical mass. XMPs are found to be outliers of the mass - and luminosity - metallicity relation, whereby they lack metals for their estimated dynamical mass and luminosity, suggesting the presence of pristine gas. However, they generally follow the luminosity - and baryonic mass Tully-Fisher relation, indicating that the H i gas is partly virialized and contains some rotational support. 60% of the XMP sources show a small velocity offset (10-40 km s-1) between the H i gas and the stellar/nebular component, implying that, in these sources, the H i gas is not tightly coupled to the stars and ionized gas. The effective yields provided by oxygen are often larger than the standard theoretical yields, suggesting that the observed H i gas is relatively metal-free. 80% of the XMP sources present asymmetric optical morphology - 60 XMPs show cometary structure, 11 show two bright star-forming knots and 18 show multiple star-forming regions. Star-formation rates are found to be similar to those typically found in BCDs. However, specific star-formation rates are high, with timescales to double their stellar mass, at the current rate, of typically less than 1 Gyr. Conclusions: XMP galaxies are among the most gas-rich objects in the local Universe. The observed H i component suggests kinematical disruption and hints at a primordial composition. Full Fig. 1, Tables 3-5 are available in electronic form at http://www.aanda.orgReduced spectra 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/558/A18