How does Positronium exist?
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.
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.
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.