Nucleosynthesis takes place in a high pressure, high temperature soup.
As for the temperature barrier, if the temperature is too low, the nucleus will be stable but will not have enough energy to form. If the temperature is too high, the nucleus can't exist and the nucleus would dissociate. If the temperature is intermediate, however, the nucleus is stable and can form. Because of a large abundance of Helium 4, we can conclude that the temperature of the universe was very hot. If enough energy is present, the potential barrier will be penetrated and a photon will be emitted.