GridFs uses the storage system of MongoDb documents to store small chunks (256K) of files. It's not an aspect of MongoDb that can be split off to be used independently.
MongoDb also, by default, creates data storage files that are 2GB. So, even when using MongoDb, the files wouldn't be stored in a single file. They would be split into multiple data files. File splits like that are not uncommon in databases.
Also, BSON is not limited to 16MB as a storage format. The 16MB limit is imposed by the MongoDb engine. BSON documents could nearly store 2GB of data per specification.