Both Manny Reyes and wikipedia are slightly off on their statement of Hochster's characterization of spectral spaces (i.e., topological spaces homeomorphic to the spectrum of some commutative ring).
Both are missing the condition that the quasi-compact opens form a base for the topology. A perhaps more transparent characterization of spectral spaces is that they are precisely the inverse limits of finite T_0 spaces (see paragraph 2 on the first page of Hochster's paper for both of these). In particular, from the latter characterization it is easy to see that a compact Hausdorff space is spectral iff it is totally disconnected (<=> zero-dimensional <=> Boolean). Thus most compact sober spaces are not spectral, e.g. the closed unit interval is not a spectral space.
Note also that the paper ends with various characterizations of the topological space of a scheme (Proposition 16): these are precisely the spaces homeomorphic to an open subset of a spectral space.