Biological signals, such as electroencephalograms (EEG), play a crucial role in numerous clinical applications, exhibiting diverse data formats and quality profiles. Current deep learning models for biosignals are typically specialized for specific datasets and clinical settings, limiting their broader applicability. Motivated by the success of large language models in text processing, we explore the development of foundational models that are trained from multiple data sources and can be fine-tuned on different downstream biosignal tasks. To overcome the unique challenges associated with biosignals of various formats, such as mismatched channels, variable sample lengths, and prevalent missing values, we propose a Biosignal Transformer (\method). The proposed \method model can enable cross-data learning with mismatched channels, variable lengths, and missing values by tokenizing diverse biosignals into unified "biosignal sentences". Specifically, we tokenize each channel into fixed-length segments containing local signal features, flattening them to form consistent "sentences". Channel embeddings and {\em relative} position embeddings are added to preserve spatio-temporal features. The \method model is versatile and applicable to various biosignal learning settings across different datasets, including joint pre-training for larger models. Comprehensive evaluations on EEG, electrocardiogram (ECG), and human activity sensory signals demonstrate that \method outperforms robust baselines in common settings and facilitates learning across multiple datasets with different formats. Use CHB-MIT seizure detection task as an example, our vanilla \method model shows 3\% improvement over baselines in balanced accuracy, and the pre-trained \method models (optimized from other data sources) can further bring up to 4\% improvements.