Anyone can build a chapel and have Mass celebrated in it as a private function. The Priest celebrating also does not have to have permission from the Bishop of the place, although he should be in good standing with his own Bishop (i.e. not suspedned, interdicted, or excommunicated). I have some wealthy friends who have done this at two of their properties in Maryland and Pennsylvania.
This is no different from having a priest over to your house and him saying Mass for you in your living room. You can even have a small group of friends attend such a Mass.
What cannot be done is to reserve the Blessed Sacrament, or have a public celebration of Mass, or solicit money during a collection as if the chapel is a parish or chapel of the Catholic Church.
Actually, this is not true.
Canon 1228. With due regard for the prescription of can. 1227 [which allows a bishop to establish a private chapel for himself], the permission of the local ordinary is required for Mass and other sacred celebrations to take place in a private chapel.This is no different from having a priest over to your house and him saying Mass for you in your living room.
This also is not generally allowed.
Can. 932, § 1. The celebration of the Eucharist is to be performed in a sacred place, unless in a particular case necessity demands otherwise; in such a case the celebration must be done in a respectable place.