The Death Star is the size of a small moon. Endor being a moon (not sure if it's small or not) has a rich atmosphere.
Could the Death Star hold a nitrogen/oxygen based atmosphere with its gravity?
(Now re-reading my question, I suppose this sounds a lot like the plot of SpaceBalls... that was not my intention)
According to various sources including the fun, but alas non-canon Death Star Technical Companion and the Star Wars: A New Hope Junior Novelisation, the radius of the Death Star was somewhere between 120km and 160km. The outer frame was comprised of quadanium steel and the interior was a mixture of quadanium steel and other metals.
Assuming the interior is completely solid metal (which it isn't) and presuming quadanium steel is similar in density to normal steel (which we don't know) then we can calculate that the Death Star has a maximum apparent gravity of around 0.04% of Earth normal.
This would be well below the point that any celestial body could retain a breathable oxygen/nitrogen atmosphere. There simply wouldn't be sufficient gravity to hold it in place.
In terms of natural gravity (the mass of the actual physical structure of the Death Star and everything on/in it, without taking into consideration any technological devices that create artificial gravity), certainly not.
The Death Star isn't even the size of a small moon in the sense that most of us think of moons (rocky objects of roughly spherical shape), by the standards of our solar system. The Death Star II was only about 100/160 kilometers miles in diameter, and the first one was even smaller. As you can see in this list of Solar System objects by size, the only objects with a diameter of 160 kilometers are asteroids, not moons (this list uses radius, which is half of the diameter of the object, so we're looking for objects with a radius between 50 and 80 kilometers). None of these objects have an atmosphere.
A moon that size wouldn't have enough mass to form a sphere, let alone build up and hold on to an atmosphere. Rocky objects that size are lumpy and randomly shaped, often referred to as "potato shaped", for reasons that I don't fully understand. And these objects are, of course, more or less solid rock, with little or no negative space inside, and relatively few cavities.
The Death Star, on the other hand, is full of negative space. It is mostly hollow, at least relative to a moon, because it has to be: it is a space station manned by hundreds of thousands of men and droids, and has living quarters, recreation areas, maintenance areas, utility areas, storage, command facilities, battle stations, etc, etc, etc.
Without artificial gravity of some kind, a mostly hollow structure like this would be incapable of holding on to an atmosphere on the surface - it simply doesn't have enough mass. If, however, it was airtight (which it presumably was) it would have to contain an artificial atmosphere, since we never see anyone inside wearing oxygen masks.
This does indeed seem to be the case. When the Star Destroyer crashes into it, we see what looks like violent outgassing (at least, that's what I think is going on), suggesting that there is little or no atmosphere on the surface, and the pressure inside is much higher than the near-vacuum of space outside.
However, this is science fiction, not the present day state of technology. We couldn't build something like the Death Star, and we can't create artificial gravity, but the people in the Galaxy Far, Far Away can do both of these things. So really, all bets are off.
The first Death Star was completed when it was destroyed, but the second one was apparently less than half finished. It was bigger than the first, but probably had less mass at the time it was destroyed, because most of the living areas and non-essential facilities were not built yet. It seemed to have only a skeleton crew, so to speak - enough men to man the battle stations and do the construction work, but not much more than that. Less men means less space needs to be airtight and filled with breathable air. Less of the outer shell being in place means the area that could be pressurized is much smaller.
One might argue that the construction teams needed air while they were building the damn thing, but I would imagine that much of the work on the outer portions of the station was performed by droids.
But in any case, since this is science fiction, they can basically do whatever the writers want or need them to do. So is it possible that the exterior of the Death Star had an atmosphere? Of course. But if it did, it was because of some sort of artificial gravity technology, not because of the natural mass of the structure itself. And I'm not sure why they would need an atmosphere outside the shell of the Death Star - I don't think anyone goes out there very often. All the good stuff is inside.
On the scientific side of the question, I found an interesting article about the conditions necessary for an astronomical object to maintain an atmosphere, as explained by a member of the Mars Pathfinder team.
Well let's consider what we need in order to have an atmosphere: Gravity. You could think of an atmosphere as a world-wide version of tug-of-war. Various forces are pushing the gases out, while gravity is pulling those same gases back in. So we would need sufficient gravity to keep an atmosphere, and in this case, size doesn't matter. We can't judge the gravity exerted by this space station just by the fact that it appears to be the size of a small moon.
The first issue at hand is that the artificial gravity within the Death Star doesn't appear to center around the Death Star. As we can see in Star Wars (Episode 4), the gravity actually appears to be perpendicular to the center of mass. With that, we cannot correlate any internal gravity with its external gravity.
Now we know that the Death Star has thermal exhaust ports to pump the excess heat from the core reactor. Based on the design and appearance of these exhaust ports, we could infer that they likely expel a heated gas. However it wouldn't make much sense to expel a heated gas if the Death Star had the kind of atmosphere that would keep it all in. Essentially, this would cook the Death Star from both inside and outside since there would be no place for this heat to go. This makes me believe that it does not maintain the gravity necessary to maintain such an atmosphere, except....
We have explosions happening on the surface of the Death Star. Depending on how much you wish to attribute to movie magic, we could probably wave this off, but let's assume it's how things actually worked. There was definitely fire resulting from these, and so we would have to assume that there was oxygen to fuel the flame. What makes this even more confusing is the fact that there are several breaches into the interior, where oxygen exists, but there is no vacuum. So either they had a way of containing these very quickly or there was an atmosphere on the exterior sufficient enough for there to be no pressure difference.
The fact that we need specifically shielded airlocks compounds this even more because we wouldn't need them if the Death Star had enough of an atmosphere to make it negligible. More than likely, this atmosphere is either partial (ie, it covers only some of the surface of the Death Star), or is held by forces weak enough for the atmosphere to be expelled at some rate.
Based on these two conflicting pieces, I would say that the Death Star does maintain an atmosphere, just not a very good one.
