Symplectic homology of the cotangent bundle is the homology of loop space (see Viterbo's "Functors and computations in Floer homology" or Abbondandolo-Schwartz).
Also, Cohen-Jones-Segal have a paper in the Floer memorial volume which outlines the construction (modulo analytical details of e.g. defining smooth structures on compactified moduli spaces) of a spectrum whose homology recovers a given Floer homology. See early work of Manolescu on the analogous problem in Seiberg-Witten-Floer homology or this paper of Lipyanskiy which extends the Viterbo-Abbondandolo-Schwartz result to the level of Floer bordism.
EDIT: I would like to say a little more about this. Cohen-Jones-Segal prove that one can construct a manifold up to homeomorphism from Morse data alone (this shouldn't be too surprising when you remember that any compact manifold admitting a Morse function with two critical points has to be homeomorphic to a sphere). So although it's true (as Steven Landsberg says) that you can construct a space by geometric realisation of a Dold-Kan construction applied to the chain complex used to define homology, it's not clear to me that this will reconstruct the original space you started with (maybe it works up to homotopy?).
The idea of Floer homotopy theory is therefore strictly deeper than just 'constructing a space whose homology gives you Floer homology'. It should really give new Floer theoretic invariants (e.g. the work of Barraud and Cornea on the 'quantum' Serre spectral sequence).