Channel charting (CC) applies dimensionality reduction to channel state information (CSI) data at the infrastructure basestation side with the goal of extracting pseudo-position information for each user. The self-supervised nature of CC enables predictive tasks that depend on user position without requiring any ground-truth position information. In this work, we focus on the practically relevant streaming CSI data scenario, in which CSI is constantly estimated. To deal with storage limitations, we develop a novel streaming CC architecture that maintains a small core CSI dataset from which the channel charts are learned. Curation of the core CSI dataset is achieved using a min-maxsimilarity criterion. Numerical validation with measured CSI data demonstrates that our method approaches the accuracy obtained from the complete CSI dataset while using only a fraction of CSI storage and avoiding catastrophic forgetting of old CSI data.