Say there were 2 objects with certain masses (e.g. $m_1$ and $m_2$). If they were close together gravity would attract the 2 objects. If they were a large distance apart the expansion of the universe (dark energy) would pull them apart. What is the distance apart that they must be (I assume that it is relative to their masses) in order for gravitational attraction and dark energy repulsion to cancel out, so that the objects remain motionless?
Unfortunately I can't add comments yet.
How much knowledge theoretical physics do you know, before I make too many assumptions (or lack thereof!)?
I don't think that this question can have an acceptable answer.
This will depend greatly on what model for the Dark Energy (DE) you are using. There are dozens, and new ones continuily being created, and we do not know which (if any) are correct.
The standard cosmology ($\Lambda$CDM) has a Cosmological Constant, $\Lambda$, which acts as the dark evergy to cause accelerating cosmic expansion today. This $\Lambda$ has to be added in by hand however, which is not very pretty. Theorists like to have everything be for a reason.
'Most' models treat the DE as a scalar field, $\phi(x)$, say, which could have any of dozens of different properties in order to give a specific DE model. I do not know too much about all of them. Usually an expansion of the potential for this field $\phi(x)$ will yield a constant term, which becomes our '$\Lambda$'.
The one that I do know something about, (and I don't want to be accidently plugging my own here, my apologies) is one in which, long story short, the interaction coupling scales inversely with the density of matter nearby.
Thus you're question of 'how far apart must two test masses be' is not relevant. The question becomes a question of 'how large must the nearby density be to screen out the $\phi(x)$ interaction.
You could have two such test masses in our Earth atmosphere which are only of the order of 1mm apart and the interaction may be neglible.
Alternatively in the vast nothingness of space and time you could have two test masses many Mpc apart and the $\phi(x)$ interaction could be large.
Even still, we don't know for sure, because we can only set bounds on these sorts of questions using current experimental results. It may be the case that there is a 'fifth interaction', or it may not. We do not know yet for certain.
Typically the scales involved for the galaxy clusters that we measure to be moving away from us at an acceleration rate are on the order of 100 Mpc or more.