Are you trying to download the staging or production environment? If your trying to download production, have you published your library at least once? The production library is not built until your first publish. This could be why you are seeing a not found error.
I know the Library Download is a hosting option, but if I'm staging the files myself, I need to download the files first. Whether or not the update occurs via POST isn't relevant. I can't download the files in the first place, because it gives me that weird file not found." response.
The Library Download is one of the 'deployment options'. It lets you self-host your DTM library files. With the Library Download hosting option, dynamic tag management sends a ping (an HTTP post) to your system to let your servers know there has been an update to the library files.
Your IT team can create a cron job, or other scripted process, so your servers can pick up the changed files. In this way, whenever a change is made in dynamic tag management, the files you are hosting on your servers can be updated automatically. Post this setup is complete, the URL is generated and it's used by DTM to make an HTTP POST call.
If you already have this setup for your DTM company, please contact your IT team who would have access to the library location within your environment and would be able to share the library files with you.