The achievement can be unlocked with any number of settlers in any settlement. Graygarden is not recommended as the happiness of Mister Handy settlers is limited to 50%.
The size of the settlement refers to the number of structures in the settlement, not the number of settlers. However, in spite of the description, the size of the settlement does not have to be large (i.e. yellow/orange color).
The happiness cap without any bonus happiness is around 80, which should be reached by providing enough water, food, beds (covered) and defense.