The ability to fabricate hafnium oxide (HfO2) nanostructures is central to the performance of computation devices, non-volatile memories, capacitors, biosensors, and optical devices. Sequential infiltration synthesis (SIS) has emerged in recent years as a new platform for nanostructure fabrication via the growth of inorganic materials within polymer templates using vapor phase precursors. However, the SIS materials library is currently still limited. In this study, we expand the SIS library with the development of hafnium oxide SIS and demonstrate HfOx nanostructure fabrication via SIS within block copolymer (BCP) templates. Insights into the infiltration behavior of tetrakis(dimethylamino)hafnium (TDMAHf), the Hf precursor, in various homopolymers, were obtained by in situ quartz crystal microgravimetry (QCM) measurements. These insights guided us to judiciously choose BCP systems for HfOx nanofabrication, where polystyrene-block-poly(epoxyisoprene) (PS-b-EPI) and HfOx SIS at 95 degrees C showed exceptional selective growth of HfOx in the EPI domains. Density functional theory (DFT) calculations supported the experimental observation that TDMAHf exhibits higher reactivity with the EPI domain compared to the PS domain. Detailed investigation of the templated HfOx nanostructures using scanning transmission electron microscopy (STEM), energy dispersive X-ray spectroscopy (EDS), and X-ray photoelectron spectroscopy (XPS) further validates the presence and distribution of Hf and O elements, as well as two oxidation states of the oxide. Our work presents the first report of HfOx SIS in BCPs, providing a method for HfOx nanostructure fabrication with tuned morphology and precise control over the HfOx growth. Furthermore, it contributes to the growing SIS library and its new avenues for functional nanomaterials with customizable properties fabrication.