The short answer is yes, it is possible to store books in the cloud and download them with the device's built-in browser. However there are some things you should be aware of:
* The browser can't navigate all cloud-storage websites, and the websites change now and then which can break things. For example,downloading currently works with DropBox, but only if you switch to desktop mode after logging in.
* There can be complications with storing the Calibre library in the cloud that can lead to the Calibre library being corrupted. You need to be careful to disable syncing of the cloud storage while Calibre is running. It is probably safer to keep the Calibre library on local storage and use the 'Save to disk' function to export books for uploading to the cloud. (Edit:
See the warning about doing this in the Calibre FAQ)
* Downloading books via wi-fi, even direct from the Calibre server, doesn't give you all the features you get when connecting to Calibre via USB. E.g. sending covers, series information, manipulating collections, and many other things only work when sending via USB.