Out of all the big cat species, only the cheetah has proved possible to domesticate to any extent. Both the ancient Egyptians and the Assyrians kept tame cheetahs and used them for hunting, as centuries later did the Mogul emperors in India.
These beautiful animals can be remarkably tractable when habituated to people—the huntsmen of old trained them to accept a collar and leash, and to return to the huntsmen when the chase was over.
However, even among those with the means to keep them, cheetahs never became widespread as pets. This is most probably because until recently people did not understand their breeding behavior well enough for them to reproduce successfully in captivity.
Replacement animals were always sourced from the wild and therefore never developed the temperament of domestic pets that are born and raised by people over many generations.