The Mystery Dungeon Series takes place in Pokemon Storage Network
The whole shebang takes place in the Pokemon Storage network inside of the main games, sometime after the original games, as early as Gold/Silver. I know only the following in support of this theory.
Bill from the main series is introduced as having turned himself into a pokemon of some kind during a teleportation device accident. The pokeballs (And PC boxes) are described to be able to make an environment the captured 'mon likes. The PC network from Bills storage system has a central server, where all the Mon's are stored (Elsewise, you could never reach them from just anywhere with a PC.)
Bill, after his teleporter incident was confused as to whether he was human or not.
So my wild mass guess:
The Player Character is sent intentionally into the PC network to check the program for errors. This explains why the levels are randomized: The infinite challenge and stimulation keeps your Mons in storage from getting bored. The Time Gears are physical manifestations of the Kernel that underpins the whole system, and the places like Treasure Town or whatnot are mass hubs on the Central Servers where the Mons are left free.
Time is also accelerated from the outsider's perspective. That's why humans and their appearance are common knowledge, but their absence is unremarked upon, as they know deep down they're in some kind of deeply involved simulation.
* Out of curiosity, what does this mean for the I Choose To Stay part of the Rescue Team games, or the ending of the Explorers set?
o As soon as I get to the ending, I will provide an answer. As for Red/Blue, the Hero lost their memory- this may have even been intentional, (Sabotage or other reasons,) at least so as to minimize the odds of getting lynched by irate mons. (Or worse yet, given a list of bugfixes.) If it wasn't intentional, then the transfer was screwed up somehow. Either way, as noted above, they can't remember much if anything of the world they were from, so they simply figure 'what the hell' and stay.