If we had accurate information about Imam Mahdi’s private life, then ‘Ghaibat’ (concealment) would had no meaning. ‘Ghaibat’ means that we have no exact information about him.
We are aware that Hazrat Mahdi (a.t.f.s.) is the immaculate Imam, and the noblest exemplar amongst all in practicing Islam. He is the embodiment of Islam. He puts into practice all the Islamic regulations within himself. Later on, he would implement these regulations world-wide. One of the Islamic regulations (and the prophet’s practice) is marriage. So Hazrat Mahdi (a.t.f.s.) surely marries and, at present, he may have children. But we don’t have any details in this regard.
A few who were honoured of visiting Hazrat have, in certain instances, seen him at some specific places, with certain people besides him. This possibility cannot be completely negated that Hazrat lives in some particular city, village, or place that is unknown to the common man. It’s also possible that Hazrat has no specific place. This too is possible that Hazrat possesses a family, spouse, and children; but they aren’t aware of his Imamate.
Suppose he possesses family and children the question arises whether they too have a long life like Hazrat? Reply: Traditions only talk about Imam’s long life and nothing is mentioned about longevity of his family members. About his children, in some Ziarat, we come across the word ‘household members’. For example, in Ziarat on Friday, we read:
“ السلام عليك صلي الله عليك و علي أهل بيتك الطيبين الطاهرين
(Bihar-ul-Anwar 12: 215)
About this topic, you may refer to ‘Discussions on Hazrat Mahdi (a.t.f.s.) : 73