My friend and I were studying for our EM test when we started to think about what happens to the electric field near an infinite line of charge.
$$E = \frac{\lambda}{2\pi\rho\epsilon_{0}}$$
As you get close to the line of charge, it seems like the electric field strength goes to infinity.
We were wondering if this means that, even in the classical picture of physics, two electrons could never touch because it would require an infinite amount of energy. Is this correct?
You are correct when you concluded that two classical point electrons could never touch each other. It would take infinite energy.
There is another 'infinity' (among others) lurking in classical electrodynamics which is evident when one calculates the electrostatic energy $W$ of a uniform spherical charge distribution of radius $a$ and total charge $Q$
$$W = \frac{3}{5}\frac{Q^2}{4\pi \epsilon_0 a}$$
Thus, by this result, a point (zero radius) particle of charge Q has 'infinite' self-energy. Even in a quantum formulation of electrodynamics, formally infinite self-energy must be addressed by a mathematical procedure that essentially 'subtracts' this infinity in order to produce finite, meaningful predictions.
That is a unusual case but it can be achieved in a particle accelerator. when they give electrons enough kinetic energy to overcome the electric force they have on each other.
keep in mind that electrons, small as they are, do have size. so for them to "touch" you will need A LOT of energy, but not infinite.
The answer to the main question is YES. Two electrons will "touch" each other when their centers are at a separation equal to one electron diameter. Since the diameter of an electron is not zero, an infinite amount of energy, is not required to make them "touch." With a (calculated) electron diameter = to $2.82 \times 10^{-15}$ m, the required energy can be calculated. Note: although a more modern calculation sets the upper limit at $10^{-22}$ m, the idea is the same.
