Complete taming of mind is possible. Buddhism teaches to detach one's mind from earthly desires so as to become free. In most literature and teachings, it shows that there were many instances where people living ordinary lives with families tamed their minds completely and opened doors to nibbhana.
This is called 'Sovahan'. It is one of the early stages of the path to nirvana (also, nibbhana). It is not easy for someone living a 'gihi' (an ordinary civil life;not a monk) to completely detach themselves from the world because of responsibilities they hold in regards to their families and the society. If your kid falls ill, a completely detached person would not worry or panic. I don't see how that would work in the context of a social being. Either it becomes hard to tame the mind in such a case or the person will leave behind that ordinary life to live an ordained life (doesn't necessarily have to be a monk).
Most people who were able to tame their minds even when living a normal life were special people during Lord Buddha's time who understood the teachings really easily once taught.
In the contemporary world, it would be hard for a person to understand and detach themselves completely due to all the stimuli from the surrounding environment. Too many distractions. This is why monks live away from their families after renouncing all their belongings.
It is a first step that forces them to live a detached life because now they do not have to care for any loved ones or worry about possessions. Now they are free to focus on meditation and taming their mind.
It is simply a fast track to detachment by reconditioning the environment to remove distractions.