We present HyperSum, an extractive summarization framework that captures both the efficiency of traditional lexical summarization and the accuracy of contemporary neural approaches. HyperSum exploits the pseudo-orthogonality that emerges when randomly initializing vectors at extremely high dimensions ("blessing of dimensionality") to construct representative and efficient sentence embeddings. Simply clustering the obtained embeddings and extracting their medoids yields competitive summaries. HyperSum often outperforms state-of-the-art summarizers -- in terms of both summary accuracy and faithfulness -- while being 10 to 100 times faster. We open-source HyperSum as a strong baseline for unsupervised extractive summarization.