The case-cohort design facilitates economical investigation of risk factors in a large survival study, with covariate data collected only from the cases and a simple random subset of the full cohort. Methods that accommodate the design have been developed for various semiparametric models, but most inference procedures are based on asymptotic distribution theory. Such inference can be cumbersome to derive and implement, and does not permit confidence band construction. While the bootstrap is an obvious alternative, it is unclear how to resample because of complications from the two-stage sampling design. We establish an equivalent sampling scheme, and propose a novel and versatile nonparametric bootstrap for robust inference with an appealingly simple single-stage resampling. Theoretical justification and numerical assessment are provided for a number of procedures under the proportional hazards model.