Q: Is it possible to save a document both locally and to iCloud

Is your MacOS X version in your signature still current (OS X Mavericks (10.9.1)?

What you want to do will be easiest with Yosemite and iOS 8 on your iPhone. Then you can store iWork documents on iCloud Drive and access them on your mobile devices, but also locally on your Mac. See this support document:   iCloud Drive FAQ

With Mavericks you can use Documents in the Cloud:  See this support document:   iCloud: About using iWork for iOS and iCloud

Does this mean that the files are stored locally as well as in the cloud, and I can gain access to the files from a cloud enabled laptop when the laptop cannot sign onto the internet?

iCloud Drive keeps a shadow copy of all files that you uploaded in your hidden user library. These will be used, when you are not connected to the Internet.

Thank you. I think I can replace drop box with the cloud.