Is it permitted for members of the faithful to set up a chapel in their private homes?
YES. And I would like to encourage this practice, for those who can.
Of course, such chapels would not have the Most Blessed Sacrament in them.
I have been blessed to know several families that were able to set up beautiful chapels in their homes. For those who have the resources and the space, I encourage this traditional practice. While the whole home should be a place where God is honored and loved, yet it can greatly aid to have a special place set aside for this reason. Many people fulfill this by having “prayer corners”, “prayer chairs”, “home altars”, and so forth. These are all good and holy things as well. But if you can set up a chapel? Even better.
The sky truly is the limit for this sort of thing. And whatever is done, should be done for the glory of God and the edification of all who will visit the special place.