We present a framework SCStory for online story discovery, that helps people digest rapidly published news article streams in real-time without human annotations. To organize news article streams into stories, existing approaches directly encode the articles and cluster them based on representation similarity. However, these methods yield noisy and inaccurate story discovery results because the generic article embeddings do not effectively reflect the story-indicative semantics in an article and cannot adapt to the rapidly evolving news article streams. SCStory employs self-supervised and continual learning with a novel idea of story-indicative adaptive modeling of news article streams. With a lightweight hierarchical embedding module that first learns sentence representations and then article representations, SCStory identifies story-relevant information of news articles and uses them to discover stories. The embedding module is continuously updated to adapt to evolving news streams with a contrastive learning objective, backed up by two unique techniques, confidence-aware memory replay and prioritized-augmentation, employed for label absence and data scarcity problems. Thorough experiments on real and the latest news data sets demonstrate that SCStory outperforms existing state-of-the-art algorithms for unsupervised online story discovery.