This is a partial answer.
There is a class of proteins called enzymes. Enzymes facilitate chemical reactions. Many enzymes facilitate in the following way: a particular small region of the enzyme has a high affinity for a particular molecule. That is, when that molecule comes in contact with that active region it tends to get stuck there. The fit is good, but the fit is slightly off in such a way that the molecule becomes deformed in a way that a particular chemical reaction occurs more readily. When that chemical reaction does occur the changed molecule has less affinity for the active region of the enzyme, and the molecule tends to float away. (This is of course a highly simplified account, for the purpose of this question.)
It has been proposed that there are classes of enzyme facilitated reactions that involve quantum tunneling.
This has been proposed, for instance, for forms of hydrogen transfer. In an experiment the hydrogen that the enzyme helps to transfer was replaced with deuterium, and then the reaction was seen to proceed a 100 times slower. If the enzymatic facilitation mechanism would be classical in nature you would not expect such a large difference, suggesting the facilitation mechanism was that the enzyme is actually creating favorable conditions for quantum tunneling.
Some general remarks:
The very fact that atoms form covalent bonds so that macromolecules such as proteins can exist cannot be accounted for in terms of classical mechanics.
But of course that is not the thrust of your question.
Many of the interactions between proteins and between proteins and other (smaller) molecules can be accounted for in terms of electrostatic force in one way or another, so all of that can be understood in terms of classical mechanics. That is, if you take as granted that atoms can bond to each other and form molecules, and you take the properties of those bonds as a given, then from that level on things can be accounted for in terms of classical mechanics
In that sense the usual form of enzymes facilitating chemical reactions can be understood in terms of classical mechanics. (But as I noted, it has been proposed that in some cases the reaction involves quantum tunneling.)
The size of the molecules is in itself not a factor. When molecules are in touch with each other then the areas of touch are small, the size of atoms, and it is at the areas of touch that anything chemical occurs
It has dawned on me that I misunderstood your question.
Given how I understand your question now I assert: the physics of quantum mechanics is a phenomenon in itself. The fact that atoms can exist arises from the nature of the phenomenon of quantum physics. The interactions of proteins with other molecules is quantummechanical in nature, just as anna v wrote.
To explain what I had in mind while writing the first version of my answer let me present a comparison.
Among the oldest scientific insights is Archimedes' principle. Archimedes principle is a forever principle because it does not depend on the nature of water on, say, the size scale of atoms.
Archimedes' principle holds good independent of whether fluids are a continuum or that fluids consist of separate molecules.
There are insights in the nature of things that remain valid even when there is a revolution in the scientific understanding.
The realisation that all matter consists of atoms was a revolution in physics understanding, but that revolution didn't affect the validity of Archimedes' principle.
Now let's take for example statistical mechanics. The development of statistical mechanics to a high level was before the introduction of quantum mechanics. Similar to the example of Archimides' principle: after the development of the science of quantum mechanics the understanding of Nature provided by the science of statistical mechanics remained as valid as before.
An example of before/after difference:
Statistical mechanics takes the sizes and interaction properties of the various atoms as a given, and takes it from there. In terms of quantum mechanics it is possible to account for the sizes and interaction properties of atoms in terms of the distribution of electrons around the nucleus.
So that is how I took your question while writing the first version of my answer