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We analyze the possibilities of detecting a signal in the hydrogen 21~cm line, which was formed in the early Universe during the the Dark Ages cosmological epoch, using the Ukrainian radio telescopes UTR-2 and GURT of the National Academy of Sciences of Ukraine. As a result of cosmological expansion, this line is shifted to the decameter range of wavelengths ($\lambda_{obs}\approx18$ m, $\nu_{obs}\approx16$ MHz) and is in the band of operational frequencies of these telescopes. The brightness temperature of the predicted sky-averaged global signal ranges from $\sim-0.08$ to $\sim0.02$ K, depending on the cosmological model. Such a faint signal is a challenge even for the world's largest radio telescope in the decameter wavelength range, UTR-2, since the signal level of the foreground synchrotron radiation of the Galaxy at these wavelengths is 20000--40000~K. The paper highlights the peculiarities of spectroscopy at the decameter waves, interfering factors of natural and instrumental origin and the ways of eliminating them in order to reliably detect the signal in the 21~cm line, which can become an important source of information both about the environment in which the first stars and galaxies were born, and about the nature of dark matter particles and the magnitude of primordial magnetic fields. It was concluded that the detection of such a signal using the most sensitive radio telescopes at the decameter wavelength range is possible (with the signal integration over the frequency band of 25~MHz), the detection time will be $\sim50$~days) and can be implemented in the coming years of peace in Ukraine., Comment: 19 pages, 16 figures; v1 - Ukrainian text, v2 - English text