Surprisingly you don't need to have that large of the pressure in the chamber to get to the sonic speed in the throat - and more than that in the diverging nozz

Surprisingly you don't need to have that large of the pressure in the chamber to get to the sonic speed in the throat - and more than that in the diverging nozzle. E.g. 3 bar pressure in the chamber could be enough for that. French Diamant rocket, the one used to launch their first satellite, had a pressure-fed first stage. Lunar Expedition Module from Apollo program had a pressure-fed ascent stage.