Neuroimaging data analysis often requires purpose-built software, which can be difficult to install and may produce different results across computing environments. Beyond being a roadblock to neuroscientists, these issues of accessibility and portability can hamper the reproducibility of neuroimaging data analysis pipelines. Here, we introduce the Neurodesk platform, which offers a sustainable, flexible solution; harnessing software containers to support a comprehensive and growing suite of neuroimaging software (https://www.neurodesk.org/). Neurodesk includes both a browser-accessible virtual desktop environment and a command line interface, mediating access to containerised neuroimaging software libraries from multiple systems; including personal computers, cloud computing, high-performance computers, and Jupyter notebooks. This community-driven, open-source platform represents a paradigm shift for neuroimaging data analysis, allowing for accessible, fully reproducible and portable data analysis pipelines, which can be redeployed in perpetuity, in any computing environment, with ease.