Reactions which are impossible theoretically Take an example of reaction like :

Burning of carbon in air to form Carbon dioxide, say heat required is 100 joule (assume so) 
And we provide the system say 20 joule heat 
Theoretically it would be not possible. But the alone C atoms present near heat source will absorb their required heat (which would be very very small indeed) and get converted into co2 
Hence, there will be small value of reactants which are converted into products then how can we say reaction is not possible. Thus at room temperature is it true that there is some concentration of products in any reaction.
If the $20$ J you mention is the average energy this will correspond to a certain temperature. However, molecules have a distribution of energies, caused by their random motion in solution or in the gas phase, so that some have far more energy than the average and some far less. The relative populations of vibrational and rotational energy levels a molecule has are determined by the Boltzmann (exponential) distribution. This means that at a given temperature some molecules are more likely to react than others. If the $100$ J you mention corresponds to a temperature at which the reaction proceeds with a reasonable rate constant then if you only supply $20$ J the rate constant will be $e^{-5} \approx 7.10^{-3}$ times smaller. This may be a small enough rate constant to suspect that the reaction is not proceeding, but its just a matter of waiting !

(If you have a temperature gradient across your sample then of course the reaction rate constant will also vary across the sample being greatest where the temperature is greatest.)