OK, maybe not. But perhaps this represents an unfulfilled need in the marketplace? You can certainly get some benefits of having a pet without having it live with you long-term (or hospitals wouldn't bother with pet therapy). And you only need to look to Japan to see the urban future:
Edit, after more reading on this fascinating topic: Apparently there was a pet rental business in the US, but it folded under pressure from animal rights groups. Pet rental is now illegal in Massachusetts, and maybe the UK and elsewhere:
Oh well... Perhaps there is still hope for cat cafes:
An animal can be damaged by an irresponsible renter- killed, traumatized, injured, de-socialized. Unlike a rental car or TV, we cannot simply throw out the broken ones (I should hope you agree with me there)
Complicating things even more, the odds that you're not going to know how to interact with a pet go way up if you don't HAVE a pet!
While animals are not people, they are similar enough to people as to be completely different from 'stuff', and if you cannot deal with the responsibility of caring for a pet I do not think you should have one at all.
(In a controlled environment that can include supervision and a responsible figure who knows what they are doing, perhaps it could work, but not if you're taking them home for a week or such)