Within Human Activity Recognition (HAR), there is an insurmountable gap between the range of activities performed in life and those that can be captured in an annotated sensor dataset used in training. Failure to properly handle unseen activities seriously undermines any HAR classifier's reliability. Additionally within HAR, not all classes are equally dissimilar, some significantly overlap or encompass other sub-activities. Based on these observations, we arrange activity classes into a structured hierarchy. From there, we propose Hi-OSCAR: a Hierarchical Open-set Classifier for Activity Recognition, that can identify known activities at state-of-the-art accuracy while simultaneously rejecting unknown activities. This not only enables open-set classification, but also allows for unknown classes to be localized to the nearest internal node, providing insight beyond a binary "known/unknown" classification. To facilitate this and future open-set HAR research, we collected a new dataset: NFI_FARED. NFI_FARED contains data from multiple subjects performing nineteen activities from a range of contexts, including daily living, commuting, and rapid movements, which is fully public and available for download.