Marc Lange discusses fields in An Introduction to the Philosophy of Physics. Fields would be useful in physics to avoid action at a distance. If one can avoid action at a distinct then one has spatiotemporal locality as he described it in Chapter 1.
However, fields are "unobservable" in the sense that one can't point to them like one can the sun or the moon. Furthermore, some physicists reject them as being real. For example, on page 42, Lange quotes Alfred O'Railly's 1965 textbook, Electromagnetic Theory:
The assertion [of the field's reality], taken by itself apart from the quantitative force-law is scientifically otiose....It is merely the physically irrelevant statement of a metaphysical conviction....This is certainly not a legitimate physical theory at all; it is the confusion of metaphysical belief with metrical physics....
A field, since it occupies all of space, could be viewed as very large.
So, to answer the question, Are there any other big unobservables?, one potential candidate would be an electromagnetic field.
Lange, M. (2002). An Introduction to the philosophy of physics: Locality, fields, energy, and mass. Blackwell Publishing.