20-Year Prospect
In the next few decades, we can probably expect to find the same number of people walking around with dogs, cats, and a schedule that doesn't allow them to be around their dogs and cats as much as they'd like to be. This is great for aspiring pet sitters (and the disposable poop bag industry) as there's really no reason to think they'll be out of a job when they're twenty years older, wiser, and more covered in pup hair.
As a, shall we say, "biology expert" of sorts, don't expect the ins and outs of the job to change much either. Food will still enter through your furry wards' front door, and exit the back. Make sure you're facilitating both processes as well as you always did, and you'll be just as much a pet sitting rock star as you were when you started the gig.