Cloud infrastructure is the collective term for all physical devices within cloud systems. Failures within the cloud infrastructure system can severely compromise the stability and availability of cloud services. Particularly, batch servers outage, which is the most fatal failure, could result in the complete unavailability of all upstream services. In this work, we focus on the batch servers outage diagnosis problem, aiming to accurately and promptly analyze the root cause of outages to facilitate troubleshooting. However, our empirical study conducted in a real industrial system indicates that it is a challenging task. Firstly, the collected single-modal coarse-grained failure monitoring data (i.e., alert, incident, or change) in the cloud infrastructure system is insufficient for a comprehensive failure profiling. Secondly, due to the intricate dependencies among devices, outages are often the cumulative result of multiple failures, but correlations between failures are difficult to ascertain. To address these problems, we propose BSODiag, an unsupervised and lightweight framework designed for diagnosing batch servers outage. BSODiag provides a global analytical perspective, thoroughly explores failures information from multi-source monitoring data, models the spatio-temporal correlations among failures, and delivers accurate and interpretable diagnostic results. Experiments conducted on the Alibaba Cloud infrastructure system show that BSODiag achieves 87.5% PR@3 and 46.3% PCR, outperforming baseline methods by 10.2% and 3.7%, respectively.