Why LH2/LOX as a rocket propellant?
LH2 has a boiling point of 20 K at 1 bar, and a critical point of 33 K and 13 bar, so it is hard to keep it liquid.
LOX has a boiling point of 90 K at 1 bar, and a critical point of 155 K and 50 bar, so it's easier.
The critical point is where the liquid-gas phase transition disappears. Above its temperature, a substance cannot be liquefied.
Also, the best exhaust velocity that one can do with it is about 4.5 km/s (
RD-0146 rocket engine; the Space Shuttle Main Engine and other LH2/LOX engines have similar values). From Tsiolkovsky's rocket equation, one can reduce one's propellant consumption by increasing one's exhaust velocity. That can be done by using an electric rocket engine. The
Dawn spacecraft 's NSTAR ion engines can do 30 km/s. A kind of coilgun called a
mass driver may be capable of at least 20 km/s.