CloudHQ is a pay service (costs a minimum of $49 a year)— with a limited, free trial — that synchronizes data between various cloud storage services [See Wikipedia]. Google Drive is natively supported on Chromebook, so CloudHQ is the missing link to pull in Dropbox data to a Chromebook.

You can keep data in sync (change in  A -> sync to B or [change in  A ->sync to B] + [change in  B ->sync to A]) between Google Drive, Dropbox, Evernote, Basecamp, SugarSync, Box, Microsoft SkyDrive etc.

When a sync pair is created in the cloudHQ service, users are given the option to choose between one-way sync or two-way sync.^[5]

One-way sync: Replication will occur from one main cloud service to a secondary account, but if files are added or revised in the secondary account, the changes will not be reflected back to the original account.

Two-way sync: All file additions and revisions made in the main account will be replicated to the secondary account.^[6] All file additions and revisions made in the secondary account will be replicated back to the main account. This creates an exact copy of data in both cloud accounts.

Alternative: The chrome app ZeroPC Cloud accomplish sync without the cost

For the price of a Chromebook plus that $50 annual fee ($150+ over just 3 years), you could have purchased a normal 11.6″ notebook that doesn’t have the limitations of Chrome OS.

An inexpensive NAS is available for notebooks, netbooks, slates and smartphones, both at home over wifi and while on-the-go using 3G/4G. Upgrading the hard drive capacity in netbook is not necessary, because 6TB is available via remote access.

http://lgponthemove.blogspot.com/2012/11/accessory-corner-buffalo-linkstation.html

Chromebook cannot access network shares (no NAS, no shared folder on another PC).