@Scott: Unfortunately, I am a bit new here.. so I am not able to insert comments until i gain 50 points, so please bear with this for a while Im about to get those points.
@Peter Clark: I dont think Hochster missed quasicompact opens forms a base for the topology, and I am sure that any spectral space is the spectrum of a commutative ring. In fact, the part with quasicompact opens forming a base is in the definition of spectral spaces, see second paragraph of the paper of Hochster. Wikipedia is the one that missed the part of the definition of spectral space (sober spaces are not enough.. original definition by Hochster never even mentioned sober spaces).
So I'd like to modify things a bit: Is a T0, sober space with quasicompact opens being a base an open subspace of a spectral space (the only thing I removed here is the condition of being compact)? I fear that if you take the alexandroff compactification of a such a space (has all property of spectral but is not compact) then finite intersection of open compacts may not necessarily mean open compact.. I havent yet given this a serious thought. But if this can work with alexandroff compactification, then I have a characterization of spaces that are a result of schemes.
Im doing a new edit.. I think I have proven the following:
A topological space the result of a scheme (by the forgetful functor) iff
It is T0, compact opens forms a basis (NOTE: by compact I mean quasicompact and not quasicompact and hausdorff), finite interesections of compact open is compact open, and every irreducible component has a generic point.
=> This side is easy.. since any scheme has this property (last proposition of hochster proves this also.. since any open subspace of a spectral space has this property)
<= if you have such a topological space then take the alexandroff compactification of it. This compactification turns out to be a spectral space (i.e. all the properties of the topological space AND compactness property). The space is an open subspace of its one point compactification which is spectral, thus an open subspace of a spectral space is the topological space that is a result of a scheme (last proposition of Hochster).