Why hydrogen and oxygen do not …

This is because  the speed of the molecules in the gaseous mixture  at room temperature does not provide enough kinetic energy  to activate the reaction during  collision between the reactants . A mixture of  gases is formed with potential to  react or explode violently  , if sufficient energy is given. to such a gaseous mixture .Even introduction of an spark to this mixture results in raised temperature among some of the hydrogen and oxygen molecules causing  their faster collision .  This , then results into an exothermic reaction proceeding violently .

2H_2(g)   +   O_{2(g)   --------->}    2 H₂ O _(g)         

The activation energy barrier to the reaction is indeed very large. The Arrhenius equation allows the rate constant, k, for a reaction to be determined: 

k = Ae^(-Ea/RT) 

where A = Arrhenius pre-exponential factor (depends on the system in question) 
Ea = activation energy barrier in J/mol 
R = 8.314 J/(K·mol) 
T = absolute temperature in K 

If Ea is sufficient large, k can be so close to zero that the reaction essentially will not occur at the temperature