As you know, Dropbox does not offer self hosted server option. So if you want to have your own private 'Dropbox' - consider Bitrix24. It's free for unlimited users and, unlike Dropbox, it’s available both in cloud and as self hosted software (you get open source access if you select this option).
Importantly, you'll be able to use Bitrix24 mobile and desktop apps when you go for on premise deployment. Assess to files can be set up according to your company structure and Bitrix24 comes with a number of document approval workflows that no one else has. And since self hosted Bitrix24 comes with CMS and open source code access, you can insert your own logo, change appearance and do much more.
* Please note that the information may have changed since the publication. For current prices and features please visit Bitrix24 pricing page.