Motivated by the recent observation of Lanzetta et al. that most luminous galaxies at low redshifts produce Lyα absorptions at impact parameter l <~ 160 h^-1^ kpc, we propose that these absorbers are clouds confined by the pressure of ambient hot gas in galactic haloes. We determine the properties of this hot gas and of the absorption systems on the basis of observational and theoretical constraints. The absorbing clouds need to be replenished on a time-scale of about one orbital time (~10^9^ yr) in the galactic halo. The pressure and temperature of the gas at radius r~100 kpc are P = (10-100) cm^-3^ K and T = 10^5.5-6.5^ K. The model requires that about 10 per cent of the gas in low-redshift galactic haloes be in the hot phase. Such gas in galactic haloes emits X-rays with a bolometric luminosity of the order of 10^37-40^1 erg s^-1^. The likelihood that such gas exists in current models of galaxy formation is discussed.