I had one memorable few hours of "empty attraction" experiences, on a day during Easter break ca. 1989, at the Magic Kingdom. It was mid-afternoon on a cloudy, gray day, and my parents decided to stay at our camp at Fort Wilderness and let my sister and me (we were 11 and 13 at the time) go to the MK by ourselves. Just after we hopped the ferry to the MK, the clouds opened up and began dumping monsoon rains. By the time we got to Main Street, we had to "swim upstream" to get to the hub, as guests poured from the MK in droves, and the park seemed to empty out completely... like magic!
We proceeded straight to Space Mountain, and rode it 5 times in a row, as a walk-on. There was nobody in the queue and no other guests visible on the attraction. We felt like rich kids who'd had the park reserved just for us!
Looking for something else nearby, we ducked into "Mission to Mars." We'd never heard of it before, but it was indoors, out of the rain, and we were the only ones in the theater. Whether that was because the attraction sucked (which it totally did) or because of the park's emptiness is anyone's guess.
We proceeded to head through the rain to Haunted Mansion (we were thoroughly soaked already so we didn't really care), but I remember very distinctly that the rain was falling so fast and hard that the drainage grates couldn't keep up. The water level was up over the tops of the curbs on Main Street -- so when we stepped off the curb and into the street to pass through the hub, we were walking in a rushing river of ankle-deep water and had to be careful not to lose our footing. We then rode Haunted Mansion a good 6 or 8 times in a row, encountering just a couple of other guests. It made the whole experience 10 times creepier, for sure!
Me and my little sister, with an empty (albeit wet) Magic Kingdom as our playground! That remains the most awesome, joyful and magical afternoon I've ever had at the MK, and I doubt it could ever be outdone.