How does Positronium exist? I've just recently heard of Positronium, an "element" with interesting properties formed by an electron and positron, and I was shocked to hear that physicists were actually working with this element, even if for a very short lifetime. I was always under the impression that matter and antimatter annihilated when they came even remotely close to each other, which is apparently not the case.
How do these two particles combine to form an element if they're oppositely charged and roughly the same mass? What kind of interactions could possibly take place before they're pulled together and annihilated?
 A: Just to add. Not only does the positronium exist, it can also interact with matter and allows you to do some interesting physics. For example, in a recent paper S. Mariazzi, P. Bettotti, R.S. Brusa, Positronium Cooling and Emission in Vacuum from Nanochannels at Cryogenic Temperature, Phys. Rev. Lett. 104, 243401 (2010) positronium created by deposition of positrons on a nanostructured surface was cooled down by collision with walls of nanochannels and thermalized(!) at about 150K. Here is a citation from the abstract of that paper:

High formation yield and a meaningful cooled fraction of positronium below room temperature were obtained by implanting positrons in a silicon target in which well-controlled oxidized nanochannels (5–8 nm in diameter) perpendicular to the surface were produced. We show that by implanting positrons at 7 keV in the target held at 150 K, about 27% of positrons form positronium that escapes into the vacuum. Around 9% of the escaped positronium is cooled by collision with the walls of nanochannels and is emitted with a Maxwellian beam at 150 K.

A: That's because they are oppositely charged that they can form a bound state: even classically you can understand that: oppositely charged charges attract each other.
While it is true that a particle and its antiparticle can annihilate each other, they first have to interact.
Positronium is a purely electromagnetic bound state: the positron and the electron will form a bound state by electromagnetic interaction (no strong interaction as they are leptons, and the weak interaction does not play a role to form the bound state).
They have the same mass, but it is not a real problem.
Quantum mechanically this problem is treated exactly the same way as the textbook example of the hydrogen atom. You first separate the centre of mass from the problem, but here as they have the same mass this cannot be neglected in the final result.
Then you calculate the interaction of one particle with the centre of mass (in the case of the H atom, this is unambiguously the interaction of the electron with the proton, but here it is one of the two lepton with the centre of mass which is in the middle).
I should also be noted that even if the bound state is stable from that point of view, the positronium will eventually annihilate because the two wave function will overlap and thus these two anti-particles can interact and annihilate.
Positronium can be formed in a variety of ways, one example, where you can create positronium in your bathroom is to have an element which is $\beta^+$ unstable. After this decay, a positron is emitted. It can then interact with the very large number of electron present in the matter and they can form a bound state: the positronium.
A: For a slightly different spin on more or less the same thing that's already been said: The idea that an electron coming sort of close to a positron will inevitably annihilate with is comes from the same sort of misconception that leads people to think that objects coming sort of close to a black hole will inevitably fall in. Neither one is inevitable, because the effect of both forces is just to pull the objects toward one another, which may or may not lead to a collision that consumes them, depending on the details of the motion of the particles before they begin interacting. 
You might be able to claim that a positron and an electron that start off perfectly at rest would annihilate without ever forming a bound state, but that's a completely unrealistic situation for a bunch of reasons, including the uncertainty principle. If they start out far apart from each other with some initial velocity, though, their fate will depend on the exact arrangement of the initial conditions. Forming a stable bound state most likely requires a third particle, as well, to conserve energy and momentum.
A: Depending on the relative spin-orientation of electron and positron, the positronium may have two spin states: if the electron and positron have anti-parallel spins (+1/2 and
 -1/2), the positronium will be of spin-singlet state with intrinsic lifetime 0.125ns and decays via two-photon annihilation. On the otherhand, if electron and positron have parallel spins, the positronium will be of spin-triplet state with intrinsic lifetime of 142ns and annihilates in three-photon mode in vacuum. However, in presence of material, the positronium has finite probability to exchange its own electron with the opposite-spin electron from the surroundings, as a result the triplet-positronium may decay in a singlet-mode (two-photon) faster than 142 ns. This mode of annihilation is called pick-off decay, which shows its importance in the appliaction in calculating the void size in any porous material. 
A: As you've noticed, it's not automatically true that a particle and its antiparticle will annihilate each other when they get close to each other. In fact, no interaction between particles is really certain to happen. Quantum mechanics (and at a higher level, quantum field theory) tells you that all these interactions happen with certain probabilities. So for instance, when a particle and its antiparticle come into close proximity, there is only a chance that they will interact within any given amount of time.
However, the longer the particles remain together, the greater the probability that they will interact and annihilate each other. This is responsible for the 142 ns lifetime of positronium as reported in the Wikipedia article: the probability of annihilation increases with time in such a way that the average lifetime of an "atom" of positronium is 142 ns.
As Cedric said, as long as the positron and electron don't annihilate each other (and remember, there is only a limited chance of that happening in any given time), they can interact in much the same way as any other charged particles, such as the proton and electron. Being bound together by the electromagnetic interaction, as in a hydrogen atom or a positronium "atom," is just one example.
