I also believe that MARV could be a great interface to a public bag exchange. It is under active development so there are more features to come that are not in the community edition (yet). We have been using their enterprise edition in my previous job and I believe the developers might be open to supporting publicly hosted instances that make the data available to everyone.
For completeness, there is also https://github.com/swri-robotics/bag-database, which is also freely available and has a similar goal to MARV, but with (I believe) a subset of features.
Also, a bag cloud storage project BotBags was announced a while back, but except from this announcements, there was nothing much yet (Announcing BotBags, the cloud rosbag storage service).
As for a central place to share bags, we would need to ensure funding for the storage and bandwidth. It seems like OSF could be the right place for that (first time I hear about it), however we would need to talk to them. I guess it only makes sense if MARV was running on their servers as well. While in principle it could be possible to run it somewhere else and access the data via their API, it probably would be way too slow.
Also, from an initial look at their offerings, it seems that the storage is not restricted currently, but the maximum file size is (“Individual files must be 5GB or less to be uploaded to OSF Storage”), which might be a problem (although splitting bags is of course possible and supported by MARV).
If we figure out where and how to host it, I’m willing to help setting up and maintaining MARV for that use case.
Maybe we could also start by hosting a public instance at some university or OSRF with limited storage to get a feel for what would be required. But if you want to get serious, that is a huge task (backups, availability, scaling storage and bandwidth, funding, …).