Consider some interacting QFT on a lattice (just to avoid infinitely large momentums). The size of the lattice is assumed to be much smaller than the size of the emergent particles (like in our world). Lets assume there are scientists living in such system and they try to measure the field (which is an "observable" by the standards of QM) at a lattice site. Would they be able to do this? Wouldn't the fact that all particles are much larger than the size of the lattice restrict probing the field at a single site? In other words would the observables in such system actually be observable (at least in theory)?

Edit. There have been two excellent answers by Chiral Anomaly and ACuriousMind, but I feel they are addressing a bit different question from what I intended. I would like to be more specific. In the world I described above, what are the restrictions on the resolution of the details of the fields the scientists can measure? For instance, if they smashed electrons faster and faster, would they get a better value of electric field near the center of an electron? Or, if they sent an electron through a non-uniform electric field would they be able to extract the value of the field in an arbitrarily small region?