How to backup internal hard drive when Mac Mini reuses to boot

I have Mac Mini mid 2011. Since I upgraded to Yosemite, my Mac Mini that never had any problems has started behaving nasty. Yesterday the login "loading bar" froze to 50% and has refused to move ahead, howsoever I try. I have tried clearing NVRAM, safe mode etc.

Finally I ran disk repair and it turned out my hard drive threw IO error.

Now I have a Mac Mini with more that 200 gb of imp data, while it refuses to boot. I have an external hard drive and an ubuntu laptop. How to I backup my mac?

Any pointers appreciated