The conventional (Zwanzig-Mountain) expressions for instantaneous elastic moduli of simple fluids predict their divergence as the limit of hard-sphere (HS) interaction is approached. However, elastic moduli of a true HS fluid are finite. Here we demonstrate that this paradox reveals the soft-to-hard-sphere crossover in fluid excitations and thermodynamics. With extensive in silico study of fluids with repulsive power-law interactions (∝r^{-n}), we locate the crossover at n≃10-20 and develop a simple and accurate model for the HS regime. The results open prospects to deal with the elasticity and related phenomena in various systems, from simple fluids to melts and glasses.