There are a lot of them. Many are not abandoned, and as is shown in the video, are stocked and used by hunters, equestrians, and hikers. (The one in the video shows a name which is easily googled.)
If you use the search feature on the homepage here on haz using "cabin az" you will find a bunch of hikes with cabins like this listed. Read the descriptions to decide if they're the kind of hike you want to embark upon! Good luck!