The Teletubbies/WMG

The entire show is a delusion by which the inmates of an oppressive prison cope with their incarceration.
Locked into a crimethink gulag, their every thought, step and action are controlled by the nigh-omniscient surveillance loudspeakers which tell them everything they must do -- such as watch "educational" propaganda films, exercise in the yard, or eat the disgusting rations of nutritional paste -- that is, "Tubby Custard" -- or waferlike "Tubby Toast." Any attempt to hide from the surveillance causes an immediate alert -- "Where have all the Teletubbies gone?" Is it any wonder the prisoners have covered all of this by replacing it in their minds with a sunny, bucolic landscape of bright cheerful colors, even unto the all-seeing infant "sun"?
 * Oh my god, this makes sense.
 * Erika's New Perfume capitalized on this, as seen [//www.deviantart.com/lance-the-young/art/The-Truth-About-Teletubbies-175932587 here].

The SunBaby is God.
He created the Teletubbies and their world purely for His amusement. All praise to the SunBaby!
 * Possibly, this means the SunBaby is an infant Haruhi Suzumiya.

The characters are all in Purgatory, atoning for their sins.
A rather pleasant-looking Purgatory, but Purgatory nonetheless.
 * You forgot -- everyone is Jesus.

The Teletubbies were once human.
But one day, they were offered a mysterious pink custard by a strange little man. It was good. They had seconds. Then thirds. Soon, they had consumed hundreds of bowls of the Pepto-Bismol like substance, reverting their intellectual capacities to those of infants and forever changing them into grotesque monsters.
 * So sort of a retelling of Odysseus' encounters with the lotus-eaters crossed with his adventures with Circe? Or essentially the first part of the fourth voyage of Sinbad now that I think of it. That does of course mean that when they're fat enough someone plans on eating them.

Willy Wonka had to test out his Three-Course-Meal Gum on someone before Violet Beauregard came along...
...and he did have all those Oompa-Loompas sitting around... And he seemed awfully blasé about the side effects on Violet, almost as if he'd seen them before, swelling up and changing colors and all that...

The Teletubbies are demons.
Sigil-shaped horns, big ugly creatures with unmoving Uncanny Valley faces, and they're presided over by a giant disembodied flaming head. Teletubbies is a Stealth Aesop that television is the work of the devil, which they couldn't show blatantly without it getting taken off the air. The robot vacuum thing is there to say "technology sucks", both in the You Suck meaning and the sucks-the-intelligence-and-humanity-out-of-you meaning.

The Teletubbies are aliens living in a post-apocalyptic world.
Everyone else has died in the war, yet the strange little 'tubbies (who may have suffered physical or mental trauma, or injuries from earlier in the war) managed to survive by living in their strange bomb shelter and now have a creepy utopia to explore full of rabbits and speakers. Any other oddities (like the baby sun or the Teletubbies' strange behaviors) are just because it's another planet.

The show is set After the End
The world appears to have recovered from a civilization-destroying nuclear winter. The teletubbies' house looks suspiciously like a bunker. The only animals present are rabbits because they reproduce so fast that they could recover quickly from an atomic war. Same applies to the only plants being grass. The "tubby custard" is an imperishable foodstuff that was stored en masse in the teletubbies' bunker in case the ground became too irradiated to grow crops. (It didn't, quite, but no one remembers how to grow them.) The teletubbies themselves are mutants, and the baby face in the sun is a cloud of nuclear dust.

Teletubbies is set in the Brave New World, uh, world.
Something went wrong with that batch of clones in the Hatchery (or they were deliberately produced that way) so instead of entering normal society, they were conditioned as rabbit handlers and housed in a special compound away from everyone else. Apart from consuming vast amounts of tubby toast and tubby custard, their main purpose in life is to stop the rabbits from chewing through any important cables or generally causing havoc. After all, even in a dystopian society you'd need someone to keep the wildlife at bay.

Television sets in the real world are actually sourced from teletubbies.
In reality, teletubbies are reared in battery farms where they don't even have room to indulge in their favourite pastime of falling over. The entire show is actually propaganda for the teletubby farming industry, designed to show teletubbies in a free range setting and thereby averting the legitimate concerns of the animal welfare lobby. (And it doesn't even touch upon the issue of genetically modifying teletubbies to produce widescreen sets, or the lucrative trade in ground-up aerials for use in eastern medicine.)
 * You have to keep them alive... barely alive.

Tubby custard is made from brains
The teletubbies eat the brains of children who watch their show and slowly brainkill them. They're modern zombies who hide their longing for brains behind a facade of cuteness. Actually, they're infecting children with zombieness, so once they were normal kids, too. The vacuum cleaner is the real villain, kind of like a vampire, sucking out their brains first, now holding them as his cute sklaves who keep children watching the show.

The house? Yeah, that's their spacecraft.
Again, the Teletubbies are aliens, possibly the last of their kind (similar to Superman), but maybe just regular colonists. They're locked out of the real controls, of course, and they just have some sort of superficial effect, perhaps training them in their actual use. It actually goes much deeper than it looks, as the loudspeakers are probably part of it. The windmill transmitters allow them to monitor intel on the humans of this world -- well, England, anyway, which is possibly the only part they're privy to -- and the Teletubbies, while they may enjoy the footage, are actually just asking to review it again so there's no mistake. The Noo-Noo merely keeps the ship clean, which is not as easy as it sounds; after all, there's no loo on board.

The Sun Baby is the viewer.
Well, not you of course. You're seeing another person watching the show.

The Teletubbies are robots that humanity left behind.
Humanity has left Earth or disappeared long ago and has left behind the Teletubbies which are robots or cyborgs (as shown by the TV screen on their stomachs) with a young child-like mentality who marvel and laugh at videos of the distant past. And all around them, nature is reclaiming the land, as shown by the half-buried structures emerging from grassy hills and rabbits running around. And the prominence of the sun implies they are solar-powered.

The teletubbies are mutants who survived the nuclear war The vestiges of humanity live deep underground.
The speakers are how the surviving humans attempt to communicate with the surface. They are hoping to control the teletubbies' actions so that the teletubbies will make Earth livable again. However, they must deal with constant frustration as the teletubbies show very few signs of intelligence. Thus, the humans are slowly attempting to condition and educate the teletubbies to understand the world without questioning it, primarily through repetition to build the teletubbies' pattern recognition.

The Teletubbies are the future of mankind after evolving significantly.

 * Eh why not? And I'm talking millions to billions of years in the future.

Teletubbies are made from people eating too many TV dinners.
I couldn't resist.

The Teletubbies live in the Midnight Channel.

 * Compare the land they live in to the True Ending of Persona 4. Ghibli Hills? Check! Televisions? Check (and in their stomachs no less)! The Sun?  Perhaps the land they live in is the collective consciousness of a country or area where the fog has been lifted?