We show that extreme vacuum pressures can be measured with current technology by detecting the photons produced by the relativistic Thomson scattering of ultraintense laser light by the electrons of the medium. We compute the amount of radiation scattered at different frequencies and angles when a Gaussian laser pulse crosses a vacuum tube and design strategies for the efficient measurement of pressure. In particular, we show that a single day experiment at a high repetition rate petawatt laser facility such as Vega, that will be operating in 2014 in Salamanca, will be sensitive, in principle, to pressures p as low as 10(-16)Pa, and will be able to provide highly reliable measurements for p >/~ 10(-14)Pa.