Briefly the answer (as to most questions about the nature of string theory) is that perturbative string theory is an *S-matrix theory*, and the space of asymptotic states (the vector space on which the S-matrix is a "matrix", namely a multilinear map) is the BRST complex of the single (super-)string. (Namely the "BRST complex", that's just the graded vector space of all excitations states of the single free string which we think of as entering into the scattering process from an asymptotic distance.) This is made nicely manifest in most accounts of string field theory. (Maybe later I find the time to write a more comprehensive reply, sorry.)