This app was mentioned in 4 comments, with an average of 3.75 upvotes
Great tutorial that I used for owncloud. It'll work with a B+ or a 2, but is much faster on a 2. I currently have a 2tb hdd attached to mine with half each for my wife and I. You can set yourself as an admin and dish out the space as you see fit.
In the comments of that link you'll find help setting it up for access outside of you network too. If that's what you want this link will help no-ip.org for dns. There's even a android app that's useful though not in Dropbox's league. There's an iOS app but I've never used it so I can't tell you if it's any good.
Edit: I've just remembered that Quickpic has recently added Owncloud support too.
I've been running Owncloud on my pi2 for a year or so. It has recently become very stable(running version 8.2.3). I used to find it had crashed once a week or so, currently it's been up for a few months without a hitch. Due to the hardware of the pi it can be slow, but for my usage it does a great job. Also the official android app has been improving a lot lately. There's an ios app but I've never used it, so I can't comment on it.
This is my go to installation guide.. It's recently been updated so it covers version 9.
The only caution I'd add in is that there has recently been a bit of a kerfuffle with Owncloud. The dude who started it has left and started up a rival called Nextcloud. But right now as Nextcloud is a fork the two are basically identical. I've managed to install Nextcloud on the pi using the same guide as above. It was just an experiment but switching the word Owncloud with Nextcloud whenever relevant worked a treat. Obviously you would be needing the Nextcloud package from the Nextcloud site rather than the Owncloud package the guide refers to. Personally right now I'd say it's probably easier to stay go Owncloud out of the two but it would be remiss of me to neglect to mention the split. It seems the community will likely follow Nextcloud but it's very early days. Nextcloud
If you plan to upload/download and generally connect from outside your network you'll need dynamic dns, I recommend installing No-ip with this tutorial..
I'll shut up now.
Just pulled it up in the Play Store from my desktop web browser: https://play.google.com/store/apps/details?id=com.owncloud.android