Structural failure time models (SFTMs) are causal models for estimating the effect of time-varying treatments on a survival outcome. The G-estimation and artificial censoring method has been proposed to estimate the model parameters in the presence of time-dependent confounding and administrative censoring. However, existing estimation method requires preprocessing data into regular-spaced (e.g. monthly) data, and the computation and inference are challenging due to the non-smoothness of artificial censoring. We propose a class of continuous-time SFTMs and semiparametric estimators for the model parameters. We show that our estimators are doubly robust in the sense that the estimators are consistent if either the treatment process is correctly specified or the failure time model is correctly specified, but not necessarily both. Moreover, we propose using inverse probability of censoring weighting to deal with dependent censoring. In contrast to artificial censoring, our weighting strategy does not introduce non-smoothness in estimation and ensures that the resampling method can be used to make inference, which is straightforward to implement in practice.