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By default, attachments, extensions, and configuration files are stored in the Confluence Home Directory that is configured when Confluence is first installed. All remaining data resides in the configured database.
Optionally, attachments can be configured to be stored in the database instead of the Confluence home directory. See Attachment Storage Configuration.
See Important Directories and Files for more information.
How can I find out how much storage space each space is taking up?
There is no direct way of retrieving the storage space due to the nature of space content being stored independently such as attachments. This certainly cannot be achieved via UI and it may be a tad tedious to search for all records in database and work out the real size of the space.
Having said this, you may try to export the space and check on the exported size.
There is an open feature request currently being tracked at CONF-3643. Feel free to cast your vote to increase its popularity and add yourself as a watcher for future updates. Thanks.