The Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey, one of three core programs of the fourth-generation Sloan Digital Sky Survey (SDSS-IV), is producing a massive, high-dimensional integral field spectroscopic data set. However, leveraging the MaNGA data set to address key questions about galaxy formation presents serious data-related challenges due to the combination of its spatially interconnected measurements and sheer volume. For each galaxy, the MaNGA pipelines produce relatively large data files to preserve the spatial correlations of the spectra and measurements, but this comes at the expense of storing the data set in coarse units or "chunks." This coarse chunking and the total volume of the data make it time-consuming to download and curate locally stored data. Thus, accessing, querying, visually exploring, and performing statistical analyses across the whole data set at a fine-grained scale is extremely challenging using just FITS files. To overcome these challenges, we have developed Marvin, a toolkit consisting of a Python package, Application Programming Interface, and web application utilizing a remote database. Marvin allows users to seamlessly work with MaNGA data by abstracting both remote and local (on-disk) interactions to behind-the-scenes data-handling functions. Combining this capability with additional processing and querying tools, users can create powerful Python workflows that are easy to import and share. Marvin's web application uses these tools to enable "point-and-click" examination of data cubes and derived maps, as well as search queries for all publicly released MaNGA galaxies. Marvin's robust and sustainable design minimizes maintenance, while facilitating user-contributed extensions such as high-level analysis code.