Gen3 is probably the closest to what you were thinking of. Self-hosting is possible, but you will probably need full-time engineer(s) in your organization to set up and maintain the system.

As soon as the amount of your data reaches a level where the benefits of having such a system outweighs the effort of setting one up, customizing and maintaining it unfortunately warrants full-time engineers. Here, a LinkedIn engineer nicely elaborates on the challenges of building and maintaining such a system. Many large companies have worked on internal tooling to make datasets discoverable across the whole organization by gathering all metadata into a central data catalogue, and basically all ended up building their own custom systems to meet their demands. Thus, there is no shortage of open-source systems you could customize to manage your metadata, but none will work out of the box and still require substantial work on your side:

• Amundsen
• Datahub
• iRods
• Marquez
• Many more...just search for "data catalogue" or "data platform"

There are also some other efforts to build data platforms with a biology/genomics focus, but as far as I know the Elixir Data Catalogue, the European Genomic Data Infrastructure (GDI) and the German Human Genome-Phenome Archive are all work in progress.

For raw data storage, you could also take a look at Hail, but maybe a simple object store with a mantis index is already sufficient for your needs. If you need to serve bioinformatic file formats via a network, various implementations of the htsget protocol (e.g. in Rust) are available. For data versioning, Restic respectively it's reimplementation Rustic could be a relatively straightforward solution that also works without much overhead on the level of single workgroups.