The production of $K_S$ and $\Lambda$ hadrons is studied in inelastic $pp$ collisions at $\sqrt{s} = 0.9$ and 7 TeV collected with the ATLAS detector at the LHC using a minimum-bias trigger. The observed distributions of transverse momentum, rapidity, and multiplicity are corrected to hadron level in a model-independent way within well defined phase-space regions. The distribution of the production ratio of $\bar{\Lambda}$ to $\Lambda$ baryons is also measured. The results are compared with various Monte Carlo simulation models. Although most of these models agree with data to within 15 percent in the $K_S$ distributions, substantial disagreements with data are found in the $\Lambda$ distributions of transverse momentum. Beam energy must be specified (in GeV) as analysis option "ENERGY" when rivet-merging samples.