Given the math of owning vs. renting, I strongly suspect they will want to begin building their own. If I were they, I'd be hiring ops developers from Google or Facebook or someone who has dealt with really big data centers and start dipping my toes into ownership immediately. They'll need to figure out how to cheaply and reliably manage petabytes of storage.
Not convinced. They are being valued at $4b not for some extra profitability by cutting costs, but because of a huge opportunity to grow. They need to do what they need for growth not hire expensive people to get into the server farm and storage business. Maybe later.
I won't argue that growth is not why they have that valuation. But, Google and Facebook, when faced with growth problems did not say, "How can we outsource our data growth problems?" They used it as a competitive advantage and a way to raise the barrier to entry for competitors. If Dropbox is paying 25%-50% more for storage than a competitor, they're going to lose as soon as a competitor comes along that gets all the other bits right.
All other things being equal, the company with lower costs wins. I'm not suggesting any other company is currently equal to Dropbox (it's what I use, and I'm constantly surprised by the number of my non-nerd friends who have the Dropbox icon on their desktop)...but, there are an awful lot of smart people out there who would love to eat their lunch.