We propose a new extra but non-cosmological explanation for fast radio bursts (FRBs) based on very young pulsars in supernova remnants. Within a few hundred years of a core-collapse supernova, the ejecta is confined within ̃1 pc, providing a high enough column density of free electrons for the observed 375-1600 pc cm-3 of dispersion measure (DM). By extrapolating a Crab-like pulsar to its infancy in an environment like that of SN 1987A, we hypothesize such an object could emit supergiant pulses sporadically which would be bright enough to be seen at a few hundred megaparsecs. We hypothesize that such supergiant pulses would preferentially occur early in the pulsar's life when the free electron density is still high, which is why we do not see large numbers of moderate DM FRBs (≲300 pc cm-3). In this scenario, Faraday rotation at the source gives rotation measures (RMs) much larger than the expected cosmological contribution. If the emission were pulsar-like, then the polarization vector could swing over the duration of the burst, which is not expected from non-rotating objects. In this model, the scattering, large DM, and commensurate RM all come from one place which is not the case for the cosmological interpretation. The model also provides testable predictions of the flux distribution and repeat rate of FRBs, and could be furthermore verified by spatial coincidence with optical supernovae of the past several decades and cross-correlation with nearby galaxy maps.