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The increasing complexity of modern high-performance computing (HPC) systems necessitates the introduction of automated and data-driven methodologies to support system administrators' effort toward increasing the system's availability. Anomaly detection is an integral part of improving the availability as it eases the system administrator's burden and reduces the time between an anomaly and its resolution. However, current state-of-the-art (SoA) approaches to anomaly detection are supervised and semi-supervised, so they require a human-labelled dataset with anomalies - this is often impractical to collect in production HPC systems. Unsupervised anomaly detection approaches based on clustering, aimed at alleviating the need for accurate anomaly data, have so far shown poor performance. In this work, we overcome these limitations by proposing RUAD, a novel Recurrent Unsupervised Anomaly Detection model. RUAD achieves better results than the current semi-supervised and unsupervised SoA approaches. This is achieved by considering temporal dependencies in the data and including long-short term memory cells in the model architecture. The proposed approach is assessed on a complete ten-month history of a Tier-0 system (Marconi100 from CINECA with 980 nodes). RUAD achieves an area under the curve (AUC) of 0.763 in semi-supervised training and an AUC of 0.767 in unsupervised training, which improves upon the SoA approach that achieves an AUC of 0.747 in semi-supervised training and an AUC of 0.734 in unsupervised training. It also vastly outperforms the current SoA unsupervised anomaly detection approach based on clustering, achieving the AUC of 0.548.