I’m not sure if this is a good replacement for Flikr, but I just ran across a new (at least new to me) open source site called Media Goblin. It is open source, so you can actually host your own media site if you have the technical proficiency and desire.
Anyway, the details can be found at http://mediagoblin.org.
Addendum Jan 18 2014: after looking at Media Goblin, I am staying with Deviant Art for hosting my artwork. Of course there are several options such as YouTube or Vimeo if you do video (which is a medium I am still trying to master) and Flikr is an option for photos and graphics. However, I’m not keen on putting my work on Flikr, so Deviant Art looks like the best solution for now. Someday Media Goblin might be a good open source alternative along with an e-commerce solution and Diaspora for self-hosting of my work, but it doesn’t seem ready for it.