I was just thinking something similar when I responded to the second post, so I've changed OP. I personally do believe that in no circumstance any private individual should be allowed unfettered access to and ownership of a nuclear device. Even if a they were able to satisfactorily protect their family and the weapon, any risk of a previously undiagnosed mental illness leading the owner to use the weapon is too great a risk.
Of course. A rule requiring multiple people's assent to grant access to the weapon is totally reasonable. Regular monitoring of mental health is reasonable too. As a practical matter, nobody is likely to go to the trouble of meeting these qualifications, so probably nobody will bother figuring them out in the first place. But in principle, there's no reason nuclear weapons shouldn't be available to civilians if they can meet those reasonable requirements needed to handle them safely.
Presumably, there is some set of requirements sufficient to ensure they're used properly and responsibly. If people can't meet those requirements, they shouldn't have nuclear weapons. If they can, why shouldn't they have them?