A quantum mechanical secret kept very well in plain sight is that the word "observable" when defined as "self-adjoint operator on a Hilbert space" does not actually mean "you can construct an apparatus that can measure this like the Born rule promises".
If you want to be formal about it, you have to consider quantum fields in the continuum as operator-valued distributions $\phi(x)$ that only yield actual operators when smeared with a test function $f(x)$ as $\phi(f) := \int f(x)\phi(x)\mathrm{d}x$.
The algebra of observables we need associate to a QFT is not the distributions $\phi(x)$, but the Haag-Kastler net of observables where on each region of spacetime $R\subset \mathbb{R}^{1,3}$ we have the algebra of operators $$ \mathcal{A}(R) = \left\{\phi(f) \mid \mathrm{supp}(f)\subset A, \phi \text{ is a quantum field}\right\},$$ where I'll wave my hands a bit and say that some expressions in the usual "fields" like their derivatives $\partial_\mu \phi$ also count as a field here. The self-adjoint operators in this algebra are the observables. (Don't try to figure out how to rigorously construct this - we do not have rigorous constructions of most quantum field theories)
For instance in QED, you can smear the electric field $F^{0i}(x)$ with a test function to get an observable $F^{0i}(f)$ that corresponds to the electric field in the region $\mathrm{supp}(f)$ weighted by the value of $f$. Whether or not this is something you can "in theory" measure depends on what sort of measurement apparati your "theory" here can construct. In the end, when the $\mathrm{supp}(f)$ gets very narrow, you'll have to concede that there probably isn't any realizable apparatus that could detect it. This, however, is not a phenomenon unique to quantum fields!
Consider an operator with unbounded continuous spectrum in ordinary quantum mechanics, such as position: You will have to admit that there is no realizable apparatus that could distinguish positions $x$ and $x+\epsilon$ for arbitrarily small $\epsilon$. But this doesn't mean we "can't measure position", it means we have to broaden our conception of measurement: What we might be really measuring is not some sharp projection onto an eigen"state" of position with eigenvalue $x_0$ (these states rigorously don't exist as states just like the QFT $\phi(x)$ is not an operator), but the projection onto some sharply but not infinitely localized state with wavefunction $\psi(x)$ centered at $x_0$ and falling off quickly far from it (how quickly depending on the details - i.e. "accuracy" - of the measurement apparatus), i.e. the projection of the original wavefunction onto some subset $X\subset\mathbb{R}^3$, but where $X$ is not a point. Note that this is morally very similar to "smearing" the position operator, just like we had to smear the quantum fields.
This was a hand-wavy description, if you are interested in formalizations of this, a more general theory of measurement considers positive operator-valued measures, and for a treatment of how to construct measurement processes for continuous observables see Ozawa's "Quantum measuring processes of continuous observables ". Notably, Ozawa proves that a measurement process for continuous observables can never be considered just as resulting in projections onto eigenstates, and does not lead to the usual property of the repeated measurement yielding exactly the same state that we are used to from discrete observables.