If there is vacuum energy, does the vacuum have mass? Assume we have a box with some volume of perfect vacuum. Does the fact that the vacuum has energy imply that the
vacuum has a mass?
As I assume, the mass of, say, deuterium gas is equivalent to
the binding energy of the electron to the nucleus, the proton 
to the neutron, the quarks of the proton and neutron - and maybe
further down. Is that right?
So, it seems like having any kind of energy in this box should make the box heavier than the box itself. Does that apply to vacuum energy? If that is the case, it seems somewhat counterintuitive.
 A: The answer to this question is really as follows: the way field theorists think of "vacuum" is not the way the layperson thinks of "vacuum." In this case, your question may have a number of answers.
If what you mean by "vacuum" is the background spacetime without particles, then that probably doesn't exist, as far as can we can observe. However, there is something called a metric field $g_{\mu\nu}$. At the level of quantum field theory, this field is generated by particles, or quanta (gravitons), which may in fact have a mass, if we could actually observe them. At this point, we have only indirect evidence of gravitons, and have not observed any at a particle collider, because gravitational interactions are quite weak. Another issue is that this view is only correct if you believe gravity is a gauge theory, which in physics pretty much means that the theory can be specified by establishing mathematical constraints on the fields. We really don't have a solid grasp on what a quantum theory of gravity should look like. For other fundamental forces, it's been possible to take the corresponding theory in the classical regime and "quantize" it (make it a quantum theory by using a strict set of rules). For gravity, this doesn't work, for a number of reasons.
On the other hand, if what you mean by "vacuum" is a set of totally empty pockets in spacetime, then I would answer your question by asking another one: can a shadow move faster than the speed of light? The answer is yes (why?). The point is that if you're looking for "empty" space, you're looking at a shadow, a lack of energy, so strictly speaking this is not going to obey the same rules as a physical object. So, I would not say it has mass.
Of course someone who studies something like non-equilibrium quantum field theory will probably say you can "excite" your vacuum -- meaning you can add energy to the natural background -- without necessarily creating massive particles. Then, the mass of these vacuum excitations is effectively zero because these excitations are purely the kinetic energy of the background field. I'm no expert on this area, but that's my best guess.
I hope that helps -- I'm trying to translate quantum field theory to a lay audience, so please let me know if anything is unclear.
A: Evan has given a nice answer based on quantum field theory, so let me give an answer based on general relativity.
The first point to note is that GR makes no distinction between mass and energy, but considers them to be equivalent and related by Einstein's famous equation $E=mc^2$.
With this in mind, the measurements made by the Planck mission revealed that the universe is flat to within experimental error, and therefore that the average energy density must be equal to the critical density. The critical density is about five hydrogen atoms per cubic metre (this is my favourite handy unit for remembering the density) and about 30% of this is matter - baryonic and dark. So there is about 3½ hydrogen atoms/m³ worth of stuff that isn't matter.
This stuff is of course dark energy. The trouble is we have no idea what this stuff/dark energy is. However we know its energy density does not change as the universe expands so it must be something that is a property of the vacuum. A plausible explanation is that it is the vacuum state of some quantum field, though I must emphasise that this is pure speculation and we have absolutely no experimental evidence to confirm or deny that.
Anyhow the bottom line is that if we exclude all the matter, baryonic and dark, we are still left with a mass/energy density equivalent to about 3½ hydrogen atoms per cubic metre. So in this sense yes the vacuum does have a mass.
However your example of the box of vacuum suggests we might be able to somehow weigh this extra mass, and of course we can do this because the vacuum is everywhere. It would be like trying to weigh air in air. The mass of the vacuum is detectable by the effect that it has on the spacetime curvature of the universe, but it is hard to see how it could ever be directly observed.
