That's no moon! It's a space station! How big can a space ship be before it collapses on itself?
Now a solid body of rock will collapse itself to a round shape when it hits about 600 km in diameter (400 km for ice). Now, the Second Death Star is estimated to be between 160 and 900 km. How big can a space ship be made of metal but still with "large open" living spaces. I assume it would still be filled with gases which would have its own gravity. Can a a spaceship be much larger than a 600 km sphere? To do so, would the infrastructure need to be primarily aluminum?
Adding from the comments.
a Dyson Sphere doesn't count, it should have internal structure.
Metal was suggested, but any material strong enough to build the self same craft is allowed.
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The defining equation of hydrostatic equilibrium - the state a celestial body must be in to maintain some semblance of a spherical shape - is $$\frac{dP}{dr}=-\frac{GM(r)\rho(r)}{r^2}$$ where $P$ is pressure, $r$ is radius, $M$ is mass, $\rho$ is density, and $G$ is the universal gravitational constant. Assuming constant density here - which is actually a problem because there are gaps - we say that $$\frac{d\rho}{dr}=0$$ and, after a quick derivation (see here for an example), we find $$P(r)=\frac{2\pi G\rho^2}{3}\left(R^2-r^2\right)$$ where $R$ is the radius of the body. At $r=0$, we have $$P(0)=\frac{2\pi GR^2\rho^2}{3}$$ Given that $$\frac{dP}{dr}<0$$ it is clear that $P$ is at a maximum at $r=0$. Re-arranging, we have $$R=\frac{1}{\rho}\sqrt{\frac{3P(0)}{2\pi G}}$$ $$$$ For $R$ to be maximized, we want the ratio $\frac{\sqrt{P(0)}}{\rho}$ to be maximized. We can say that $P(0)$ is the ultimate compression strength of a material.
Let's take a look at the strengths of various materials. The metal with the highest ratio is pre-stressed steel, at $$R=\frac{\sqrt{3,757,000,000}}{{1440}}\cdot\sqrt{\frac{3}{2\pi G}}=3,600\text{ kilometers}$$ That sounds pretty good to me.
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