The 5703WA from Raytheon, is a post WWII experimental type.
Some other subminiature valves in the '57' range are clearly for military purposes (Type 5768 is listed as a 'rocket triode'! Probably designed for use in proximity fuses.).
One advantage of the very small size and weight is robustness to vibration and shock. Such a small valve would get very hot at the ratings indicated but it might not be required to operate for more than a few seconds.
The thin glass tube envelope is 9 mm in diameter and, excluding the base pins, is 38 mm tall.
Absolute Maximum Operating Conditions¶
Updated December05, 2012. shape:thin.glass.tube construction:all.glass type:triode base: pin: