Is there a system similar to Bitbucket which I could self host? I\'ve tried to look around in the net to see if there was something but I can\'t seem to find any. We\'re using R
BitBucket is very nice, but it is not available as download since Atlassian aquired the team.
I'm not sure if you consider commercial products, but Kiln and CodeBeamer can be options to explore:
Issue tracking, wiki, etc. are out of Kiln's scope, so you will need to keep your Redmine as well, what may or may not be an advantage.
The features you mentioned (multiple repo per project, forking, pull requests) are supported by CodeBeamer, plus it is able to replace your Redmine instance completely.
(Disclaimer: Kiln is a FogCreek product, while CodeBeamer is a commercial software developed by our company)