The coulombic force between two charges is $$ F = \frac{k q_{1} q_{2}}{r^{2}} $$
For two negative charges this will be repulsive. From the equation, as $r$ tends to zero the force approaches infinity. Which implies an infinite amount of energy is needed to bring the two particles together.
My question is, how then particle accelerators manage to collide two equal charges? Is it so it can't be explained with classical mechanics?
"Colliding" two particles does not mean to bring two classical point-like particles on top of each other.
Even a macroscopic, Newtonian collision of two spheres does not consist of the two spheres somehow occupying the same space - their circumferences "touch", and upon trying to move into each other they repel each other and then separate again. See this question and this question for more discussions what "touching" means, and how things repel each other, but in essence "touching" just means that they get as close as they can under the presence of the repulsive potential between them, and then get pushed apart again.
Now, quantum mechanics makes everything more complicated because particles are not little point-like balls but quantum objects that do not possess a definite position, and the quantum mechanical notion of "shape" is rather different from our classical intuiton, see this question. It is not clear what concepts such as "touching" would mean for objects without a definite position, and so there is no really classical-like description of what happens during such a scattering.
For "slow" particles, a quantum mechanical computation called the Born approximation is typically applied to describe how particles scatter off a target with a potential.
At high energies in colliders, quantum field theoretical processes become more relevant that not only allow for mere changes of direction of the incident particle, but also for the creation of a shower of new particles. The computation of scattering amplitudes that describe the likelihood of different outcomes of such collisions is an essential part of quantum field theory and one of the major experimental tests of our current theories like the Standard Model.
This comes down to a question of what it means for particles to collide. This seems obvious because we know that two billiard balls can collide, but actually even for two macroscopic objects like billiard balls what it means for two objects to touch each other is surprisingly complicated. For more on this have a look at the question What does it mean for two objects to "touch"?
Imagine we fire the two electrons directly at each other i.e. those is a head on collision. As the electrons approach each other their mutual repulsion will slow them down until eventually they come to a halt, then accelerate away from each other again. To the experimenter watching in their lab this looks just like a collision i.e. the electrons came together, hit each other, then bounced apart again. You can argue that the electrons never touched each other, but as discussed in the question I mentioned above "touching" is a concept that only applies at a macroscopic level and fundamental particles never "touch" each other in the usual sense of the word.
So the issue you raise in your question is absolutely spot on. It's just that for elementary particles in a collider a "collision" just means the particles came close enough to interact.
There is another aspect of this that is worth mentioning because it is directly relevant to what colliders are used for. If we take the equation for the potential energy of two electrons:
$$ U = \frac{ke^2}{r} $$
then a bit of quick arithmetic tells that if we push the two electrons to within about one femtometre ($10^{-15}\mathrm{m}$) of each other then the energy reaches $10^6$ electron volts. But this is enough energy to create an electron-positron pair i.e. the energy of the electron and positron calculated using Einstein's famous equation $E = mc^2$ is about $1\textrm{MeV}$. And this is exactly what happens. If you collide electrons at energies above $1\textrm{MeV}$ some of the energy in the collision goes into creating new particles such as electrons and positrons, or even Higgs bosons if you use enough energy.
This particle production means the interaction between the particles at close range becomes far more complicated than the simple Coulomb force law would predict.