How can one overcome the idea of fornication and having a private home, despite the parents' rejection?

If by "family" you mean your parents, then honoring and obeying them is obligatory. You should try to convince them of the idea of independence. Know that independence in housing is not a condition for marriage unless the mixing of your wife with your non-mahram (definition) brothers leads to something unlawful, in which case you must inform them of the necessity of separation. However, if your parents prevent you from marrying and you fear hardship, then you must proceed with marriage.

If by "family" you mean your siblings, then it is preferable that you strive for what pleases them, but it is not obligatory upon you to do so. Try to discuss with them to convince them or to relinquish your idea. If your separation leads to the neglect of your sisters, then it is better to live with them. However, if they are males and females and there is someone to take care of them, then you are not required to stay with them; rather, it is permissible to live independently. In fact, it becomes necessary if staying with siblings leads to mixing with non-mahrams, due to the Prophet, peace and blessings be upon him, saying: "Beware of entering upon women." They said: "O Messenger of Allah! What about the in-law?" He said: "The in-law is death."