The nearest would still be hundreds of light years away on average, way too far to hurt us (they'd have to be very roughly a hundred light years or closer before the explosion would affect us), but close enough to spot in surveys. According to the paper authors (PDF), there may be thousands of these systems in our Milky Way alone. The hypothesis predicts any supercritical white dwarfs must be spinning pretty dang fast, which would be detectable. It may also be possible to measure how rapidly the star is rotating by looking for a Doppler shift in its spectrum the shift happens as one side of the star spins toward us and the other spins away. If one of them is 1.5 times the mass of the Sun - more than enough to explode - we have a winner. Because they're in binaries, we can use the orbital period to get the masses of the two stars (using math Newton invented four centuries ago!). If this idea is correct, we might be able to find such stars. The stream of hydrogen onto the first white dwarf would've cut off long before, so we don't detect it. But the delayed time bomb scenario may fix that if it takes a billion years for the dwarf to slow its spin, then by that time the other star may also have expelled all its outer layers, evolving into a white dwarf itself. But we should see some hydrogen, since the other star is dumping it onto the white dwarf. The explodey white dwarf supernova is characterized by a lack of hydrogen in it (the other kind of supernova, when the core collapses in a massive star, is lousy with hydrogen since the star's outer layers are loaded with it). The research I mentioned at the top of this post was theoretical - it's hard to get a white dwarf into the lab - but it does explain a pesky problem we've been having. Fusion of the material begins, and BANG! Supernova. At some point - and this may take a billion years - the white dwarf slows to the point where centrifugal force can no longer win the fight against gravity. Various factors slow the star down over time (for example, a magnetic field will accelerate particles in the stellar wind, acting a bit like a parachute dragging on the white dwarf). So what you get is a white dwarf with more than enough mass to explode, but its spin prevents the supernova from occurring. But if it spins really fast, then the centrifugal force acts against the force of gravity, supporting the material *. This infalling matter can then make the white dwarf spin faster. As the material falls from the red giant onto the white dwarf, it tends to spiral in due to angular momentum - the same idea of how an ice skater spins faster when they bring their arms in. The white dwarf tears itself apart, and you get one of the biggest and most violent explosions in the Universe: a supernova. Other times, if enough matter piles up - making the total mass of the white dwarf a bit more than 1.4 times that of the Sun - the ignition of fusion can cause a runaway reaction in the star, disrupting it entirely. Sometimes the material explodes, flaring in brightness, and we get a nova. If enough mass piles up, the immense gravity of the dwarf can induce nuclear fusion. It too eventually swells up, and can start to dump matter onto the dwarf (like in the picture above). It's made even uneasier by the other star.
They have incredibly strong gravity, which wants to crush them down even further, but they are supported by the electric repulsion of electrons, which is a pretty mighty force. The size of the Earth but with the mass of a star, white dwarfs are pretty weird. After a few millions years, all that's left is its core: a dense, hot ball called a white dwarf. One star nears the end of its life, swells up into a red giant, and blows off its outer layers.
Imagine a binary system of two stars like the Sun, orbiting each other. But it's still a very cool scientific question, and actually a fairly simple concept. If these things were that volatile we wouldn't be here to talk about them in the first place. Let me be clear, as I always must be when covering topics like this: we're not in any real danger from these things. Some new research just released asks a question near and dear to me: are there thousands of spinning white dwarfs in our galaxy, just waiting to explode as they gradually slow their rotation? The answer is very probably yes.