For starters, owls cannot be kept inside of a cage. They must be tethered to a large perch using a special leash when they are not being directly attended to, and due to the strong bonds they form with their owners, this cannot be for more than a few minutes at a time. Unlike other types of large birds like macaws or cockatoos, owls do not live in flocks in the wild, making them extremely antisocial with other birds except for their mates and their offspring.
The flock mentality is what allows a parrot to successfully integrate itself into a human family. Since owls lack that sort of mindset, they view everyone except for the one person that they choose as their "mate" as an enemy, and will likely attack others on sight. This can be a huge problem if you ever become sick or otherwise unable to care for your owl -- and since they mate for life, they may become depressed to the point of death if they are transferred to another person's care.
Diet is another concern that makes it nearly impossible to successfully keep owls as pets. Since owls are technically a bird of prey, they have specialized nutritional needs that cannot be met by feeding them seeds, pellets, or fresh foods like fruits and veggies. As a matter of fact, if you presented any of these treats to an owl, he or she would likely not even recognize them as food! Owls are carnivorous creatures, and must be fed whole rodents in order to meet their complex dietary needs. As one might imagine, this can be a less than pleasant ordeal to witness, much less to have to take part in several times per day. Like all birds, owls can be quite messy eaters, so cleaning up after meals is another chore that most people would find hard to stomach.
As meat eaters, owls are equipped with a beak and talons that are specifically designed for ripping and tearing chunks of meat -- and they are capable of doing quite a bit of damage to human owners if they become displeased for whatever reason. Because of this and a plethora of other reasons, it is illegal to keep owls in many areas of the world, which is another thing that makes it difficult to have one as a pet. Those who choose to defy the law and keep an owl anyway face a variety of additional problems -- what happens if the bird gets sick? Taking an owl to a regular veterinarian puts an illegal owner at risk of being caught, fined, and possibly jailed. If you are interested in a pet owl, please do yourself a favor and consider adopting a pet parrot instead. They are just as beautiful and intelligent as an owl, they eat a reasonable diet, and they are far easier to care for!