Time fields, or cracks in the space of time. Cosmologists have claimed there might be "defects in spacetime" called cosmic string -- basically tiny cracks in the universe. But don't worry; they're just microscopic enough to not fall through.
Physicist Benjamin Tippett recently wrote a white paper with colleague David Tsang about how a real-life Tardis would work. "Transversable Achronal Retrograde Domain in Spacetime" is a better name for it apparently, but for the layman, it means something that can go backward in time and move faster than the speed of light. Which is fundamentally what the Doctor's ship does. Tippett's version, however, can't go through walls. Lame.
In the "Doctor Who" episode "Turn Left," the planet is in danger of losing all of its stars. This actually is happening in our universe, though at a much slower pace. Eventually all stars explode and die. Whenever a star does go out, it takes light years for us to find out here on Earth.
Companion Donna Noble tries convincing the 10th Doctor that the bees are all going back to their alien planet. Here on the real Earth, colonies of honey bees have been disappearing. The epidemic is called "colony collapse disorder" by scientists, and the cause is still unknown.
You'll be delighted to know that we all have this remarkable capability. We are constantly regrowing scar tissue over wounds, our bones knitting themselves back together, and blood flow stopping itself. We just aren't as fast about it as the Gallifreyans are. Apparently frogs and insects can be revived after being frozen solid, and flat-worms can be cut in two to become two living flat-worms.
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There actually are sonic screwdrivers!
The famous "Doctor Who" gadget has saved the Doctor's life on many occasions, and now real-life doctors are using a similar gadget to save lives too.
Luckily, we have yet to be accosted by killer stone angels who move only when we're not looking.
That said, there are, apparently, stone monuments that have been replicated in uranium, and they have an uncanny habit. According to the Quantum Zeno Effect, uranium doesn’t decay while being observed -- only when people aren't looking. Researchers at the University of Texas discovered this effect in 1977 before there ever was such a thing as a weeping angel.
The idea of cryonics comes up in 1975's episode "An Ark in Space." Cryonics – basically freezing an organism at a low temperature to preserve it for a future period -- is still in its experimental stage among scientists, but...
...it's a largely accepted idea that the personality and memories of a human could survive within a cryonically frozen brain and emerge unharmed upon resuscitation. In the meantime, people are already arranging to cryogenically preserve themselves after death, via the American Cryonics Society and other groups.
In a 1983 episode, an antimatter entity (aka Omega) crosses into normal space via the "arc of infinity" and tries to bond with The Doctor. This could technically happen.
Quark stars -- stars made up entirely of quark matter -- are theoretical objects studied by astrophysicists. The idea is that quark stars would be highly magnetized and would consist of unstable "strange matter" that may try to latch on to more stable entities in our universe.
The 1964 episode "Planet of the Giants" aired practically in tandem with Rachel Carson's novel "Silent Spring," about a deadly insecticide called DN6. In the episode, the Tardis gang shrinks to the size of insects and explore the dangers of pesticides, which, as it turns out, are just as dangerous as the Doctor feared.
In the 2005 two-part adventure called "The Empty Child/The Doctor Dances," Rose and The Doctor head to WWII London, where nano-genes -- legions of tiny robots designed to heal injuries -- are wreaking havoc.
Nanotechnology, the science of matter manipulation on a atomic/molecular scale, is very real, particularly the field of passive nano-materials. In fact, we've been using nano-tech in items as mundane as golf balls for more than a decade.