The bad news:
You can't using the REST API.
The good news:
You can using XML-RPC, which is like old REST API.
See Remote Confluence method under Spaces -> Retrieval:
String exportSpace(String token, String spaceKey, String exportType) - exports a space and returns a String holding the URL for the download. The export type argument indicates whether or not to export in XML or HTML format - use "TYPE_XML" or "TYPE_HTML" respectively. (Note: In Confluence 3.0, the remote API specification for PDF exports changed. You can no longer use this 'exportSpace' method to export a space to PDF. Please refer to Remote API Specification for PDF Export for current remote API details on this feature.)
Oh, and disregard the token parameter. Pretend it doesn't exist.