That said, perhaps DropBox could sell a self-hosted version of their software and bring over their ease-of-use.
That's already been done.
The challenge DropBox faces with a self-hosted iteration of its software is that it stops being 'simple'. Existing Dropbox clients would have to be completely rewritten to go from asking "username and password, please" to "username, password, server address, and port, please". Even if we hand-wave away that problem by assuming that users can either correctly type a server name and port number, or that Dropbox will still have 'accounts' but essentially become a DynDNS clone and simply handle network traversal and matching users to their data repositories, we then have to deal with the Dropbox Server software. There may be a market for Dropbox to sell drives like these, but I don't see Western Digital wanting to partner with Dropbox to provide redundant functionality to their existing apps, and I don't see consumers paying more for a Dropbox branded drive if they're already in the "self-contained NAS" market - a handful might, but now Dropbox, for all intents and purposes, finds itself with all the challenges of being an external hard drive vendor...with the added bonus of directly competing with the vendors from whom they're sourcing their parts.
The obvious alternative to this would be for them to sell their software and let it run on a LAMP/WAMP stack, on whatever hardware is on hand, and market it to the enthusiast/enterprise market, like UnRAID or Nexenta. That might be a short term win, especially if they do some fancy stuff with LDAP/Active Directory integration. Conversely, I see it potentially being a support nightmare based on how it deals with storage. Will it install on an Ubuntu desktop containing a hodgepodge of hard disks? Would it be more like FreeNAS where it makes its own software RAID, but requires hardware to be dedicated (or its own VM)? Even at that, how do they bill for the software? One-time use seems like it wouldn't be a good long-term plan, but I don't see too many users being okay with Dropbox charging them an annual fee to use their own hard drives. CALs could be a useful method (arguably the most workable one), but they'd have a hard time managing their consumer-friendly image on one hand with Oracle-style licensing on the other.
Levie is right; 'free' isn't a business model. Dropbox's 2GB number is only sustainable because they're betting that a certain number of those users will go for a paid tier. Either every Dropbox customer will pay, or they start advertising, or they data mine. To my knowledge, those are the three business models that have sustained themselves on the internet. 'Everyone Pays' may be a viable model if Dropbox can do things like sell gift cards for their service (for users unable/unwilling to fork over their Mastercard) and come up with the right formula of how much customers are really willing to pay for storage+ubiquity+simplicity. Although Levie must certainly be feeling the pinch from Microsoft's 1TB of OneDrive for $60/year, the one client we attempted to migrate to that service went back to dropbox VERY quickly because the desktop client was utter crap; I'm left to believe that Dropbox's simplicity still has an edge just yet. Conversely, I don't think that $50/month for 500GB is worthwhile, either - That's only slightly less than it'd cost to buy a 500GB hard disk outright from Newegg every month.
Dropbox is still a well-recognized brand that I'm certain many consumers are still willing to pay a premium for, and Microsoft and Google are competing not only with more storage for less money, but with integration as well - editing a spreadsheet in Sheets or Excel and seamless saving of attachments is not the kind of thing that Dropbox can effectively compete with. Dropbox's best bet right now, in my half-asleep opinion, is to see how much value-add they CAN provide to their existing tiers. I can't quite fathom what that is (a trivial example off the top of my head would be an IM client add-on), but one thing is for sure: they can't easily compete against companies who sell their own gigabytes by selling someone else's gigabytes.