The energy(density) of an electromagnetic wave is described by the Poynting vector. This is the function of the $E$ and $B$ field. (the exact formula is well known). The magnitude of the $E$ and $B$ vector oscillates as the wave propagates - at some point both of them reaches its maximum (so $S=max$), at some point both of them are zero (so $S=0$). The question is: where is the energy 'stored', when both these vectors are zero?
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The Poynting vector is not the energy (density) in the electromagnetic fields, which would have units of J m$^{-3}$, it is the flux of energy per unit time with units of W m$^{-2}$.
That the Poynting vector is (instantaneously) zero tells you that there is (instantaneously) no flux of energy per unit time carried by the electromagnetic fields.
As you correctly point out, this is because the E- and B-fields are simultaneously zero (for a TEM wave) and it is the magnitude of the E- and B-fields which give the energy density $u = 0.5(\epsilon E^2 +B^2/\mu)$. Where has the energy gone? Everywhere else. The E- and B-fields are only instantaneously zero at that point in space, they are not zero everywhere. The energy can be moved around, which is what the Poynting vector and its divergence) are telling you about.
ProfRob
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