Can two quantum systems ever NOT interact? Given that two quantum systems will always be connected by fields, is it really possible for two quantum systems to remain completely unentangled? Even if the entanglement is destroyed by observation, won't the entanglement immediately creep back in?
 A: 
Given that two quantum systems will always be connected by fields, is it really possible for two quantum systems to remain completely unentangled?

In the abstract: no, it is not possible.
In practice: entanglement is not really a black-and-white dichotomy, and the degree of entanglement matters a great deal. (Suitable measures are the purity of the state of the system, the entanglement entropy, and the entanglement spectrum, among others.) If you have a system which is in-principle in a pure state, but it has some unwanted van-der-Waals interactions with a hydrogen atom on the Moon which takes the purity from $\mathrm{Tr}(\rho^2)=1$ down to $\mathrm{Tr}(\rho^2)=1-10^{-10^{10}}$, does it really matter that the system is formally not factorizable? (particularly if you take into account that no practical measurement would ever be able to detect it.)
As a general rule, there are extremely few effects where the difference between a "truly" pure state, and a mixed state which is "$\epsilon$ away" from a pure state, actually makes any real difference; to the extent that such schemes exist, they are generally (and rightly) regarded with suspicion, as they represent an intolerance to noise and experimental uncertainty that is incompatible with practical reality.

Edit: As pointed out in the comments, this reasoning holds only if you want the system of interest to be in a pure state. It is possible to protect against entanglement (with some pre-specified environment) by placing your system in a mixed state. But, particularly if you start from a pure state, this can only be done by bringing in some ancilla system, entangling it with your system, and then discarding your ancilla.
(It is an open question, currently down to unanswerable questions of quantum interpretation, whether it is possible to ever have a "truly mixed" state, i.e. a system in a mixed state which is not in that condition because it is actually in a globally-pure entangled state with some other system out there which we're unable to identify.)
