Humans have indeed caged lions. Since at least from the time of the Egyptians, lions have been in organized captivity.
Excluding the individual personality of an animal, it is my personal belief that any feline species CAN be "tamed", however, this does NOT in any way mean that it is not difficult nor dangerous. The general rule seems to me to be the larger the kitty, the more likely that natural instincts such as chasing, pouncing, and killing can be accidentally triggered. The following true story [if I recall it correctly] serves to reinforce my "layman" understanding of this animal (as I have no "hands on" experience with a big cat):
A lion raised by humans once attacked and pinned a keeper that stumbled and fell to the ground. The cat held him there with a bite to the back of the neck. Once trapped, this man wisely did not resist the cat, as this would have only provoked more aggression. The first attempts to drive the lion off the man using sticks, boards, other blows to the lion failed. It wasn't until someone lifted a full sheet of plywood and charged the cat, did the lion release the man. Luckily, the lion had not fatally injured the man.
I assume the lion perceived this wooden "object" as threatening and is similar to the "respect" elephants (as well as rhinos and hippos) get from lions in the wild. In the case of a house cat, a human is obviously already significantly larger than the feline and therefore the cat has little chance to view a person as a possible prey or "play toy".
I understand the desire to own a big cat. But I for myself, I for a start would have to be able to purchase one square mile of open grassland. I believe in giving such an animal plenty of roaming space. I would also have to be able to afford the vast cost of supplies, have access to a qualified veterinarian, and (especially in the case of lions) have two animals (after all, they are the most social wild cat).
If a person is brave enough to want to be able to physically interact with a wild cat, considerable time must be invested in developing a rapport with the animal. Ideally, the relationship should start when the feline is a cub and must be continually maintained. To accomplish this, one must be willing and able to donate most of their time to the task.
Few people have the necessary resources and stamina to accomplish taking care of such an animal, and as a result, most these wild cats end up in the rare rescue facilities that are able to house such a creature. I highly recommend that if a person can't go without experiencing an interaction with a wild cat, to locate a rescue facility or a zoo and donate your time to assist in the care of the animals. If rules of the establishment and laws permit (which in the US, varies from state to state), and other factors, the animal handlers might allow physical interaction. Always follow the keepers' instructions and always remember the feline's instincts are always present. Keep in mind that even the experienced and qualified humans can get maimed or even killed by a momentary lapse of judgment or an instant of appearing vulnerable. Lastly, even a good-natured greeting of a big cat can injure an adult man.
For those people who can admire a lion from a distance, the are plenty of zoos that keep lions. Lions are fairly easy to maintain in zoos and other large scale facilities, although not til recently has the accommodations been routinely humane. Although lions make a popular exhibit, most small manageries and/or those of low funding opt not to house big cats due to the high cost of upkeep.
