When people discuss quantum field theory in a popular context, they say that fundamental particles, such as quarks and electrons, are pointlike, with zero size.
However, I don't think this is what QFT says at all. My impression is that a particle is an excitation of the field, and hence a wavepacket with finite size. Of course, you can have a spike in field value at a single point, like $|x \rangle$ in quantum mechanics, but this state is pathological, and it's definitely not what we mean when we say 'particle'.
Interactions between fields are local and hence only happen at a single point. But to calculate the result of a scattering, you have to integrate this over the positions where the incoming wavepackets overlap. So it doesn't make sense to say that interactions occur at a point either.
If field excitations are not pointlike, and interactions of fields don't occur at points, then what do people mean when they say particles are pointlike?
Edit: I don't think this question answers mine. The top voted answer just reiterates that the Lagrangian is local, but most of the time people seem to be making the much stronger statement that particles are perfectly localized. As argued above, these are not the same thing. I'm wondering if there's any justification for the latter, or if it actually doesn't make sense.