In this paper, we introduce the MoisesDB dataset for musical source separation. It consists of 240 tracks from 45 artists, covering twelve musical genres. For each song, we provide its individual audio sources, organized in a two-level hierarchical taxonomy of stems. This will facilitate building and evaluating fine-grained source separation systems that go beyond the limitation of using four stems (drums, bass, other, and vocals) due to lack of data. To facilitate the adoption of this dataset, we publish an easy-to-use Python library to download, process and use MoisesDB. Alongside a thorough documentation and analysis of the dataset contents, this work provides baseline results for open-source separation models for varying separation granularities (four, five, and six stems), and discuss their results.