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