We analyze the possibility of detection of cosmological relic neutrinos via neutrino capture on beta decaying nuclei. This reaction has no threshold in neutrino energy, which is crucial for searching for relic neutrinos possessing very low energies. We focus on tritium ((3)H) and rhenium ((187)Re) beta radioactive isotopes to be used in the Karlsruhe tritium neutrino mass (KATRIN) experiment and project, called the Microcalorimeter Arrays for a Rhenium Experiment (MARE), dedicated to the measurement of the electron neutrino mass at sub-eV scale. We examine these experiments from the viewpoint of searching for the cosmological neutrinos via neutrino capture. We conclude that even with possible gravitational clustering of relic neutrinos the prospects for their detection in these and other similar experiments are not optimistic. Nevertheless, KATRIN and MARE experiments could establish some useful constraints on the density of relic neutrinos.