Of course, I have to take a different route to answer this as I guess I looked at the original post through survival-tinted lenses. The following statements relate directly to a survival situation, not otherwise legal harvesting of game.
First, like some, I would recommend a dedicated firearm for each size of animal. I do know of people who have taken (illegally) white-tail deer with .22 magnums. I, myself, have taken rabbits with the .22 mag, but would not take anything much larger than, say, a raccoon with it.
Similarly, I've taken large game, including deer, with .308, .30-06 and .30-30. I could take rabbit with it, but they'd have to be head shots and, even then, that'd be, if you'll pardon the term, overkill. . . an inefficient use of resources.
As ChrisFox suggested, I'd most likely recommend a 12-guage shotgun due to it's versatility. One gun using different types of ammo should address your problem.
However, I would also recommend the Savage 24 combination gun, under certain circumstances. I have a couple combinations of these. one is a .22 magnum over a .410 shotgun. The .410 shotgun could be used to take deer with buckshot. I would recommend against it in all but the most dire situations. I've used the heck out of this combo gun and wouldn't have a problem taking it anywhere but Alaska as a survival/subsistence gun if a selection of .410 ammo were carried (i.e., birdshot and buckshot).
Finding one of the .22 magnum/.410 combos is difficult. It was a short run through, I believe, Sears. . .when they sold such things. The more common .22 LR/.410 shotgun combo is more easily found.
What I'd recommend, but haven't seen with my own eyes, would be a .22 (LR or Magnum) over 12-guage. This would not only allow me to take small and large game, but offer protection against bear.
Another variation is the .30-30/12 guage combo. IMHO, this would be a less-than-ideal combo. Mainly because despite the rifle caliber being able to take large game, the shotgun is really too small to take bird or small game unless you got very close. To do so would require a modicum of hunting/stalking skills and experience, something which is not indicated here.
And, taking a quick sojourn into Fantasyland, there is the option of getting a custom-made combination/drilling gun. I'd recommend a 24" side-by-side 12-guage with one barrel having either a rifled or slug bore and the other with either a modified choke or threaded to take choke-tubes. Centered above this, I'd put a .22 LR barrel.
Sadly, this wouldn't be optimized for anything. The rifled slugs would do well against large animals and the .22 would do pretty good for small game. The smooth-bore barrel would be too short for most anything but birds flushed at close- to medium-ranges and ground-based birds such as the turkey.
Probably not what you wanted to hear, but there it is in a survival-based format.
The Professor