http://www.universityofcalifornia.edu/infocenter/freshman-admissions-summary
You can use this tool to see the GPA breakdown by UC campus. Unfortunately, it shows just 1% of admissions had between a 3.0 and 3.4 at UCSB last year.
It isn’t impossible but it it highly unlikely. Apply anyhow and explain your situation in a compelling manner in your essay. I hope it works out for you.
At the same time, recognize your admission to any UC straight out of high school is really unlikely due to your GPA. If you want to attend a UC, go to a CC and transfer. You are obviously bright but, circumstance didn’t favor you in high school. Bust your ass, get a 3.5 and the TAG framework to transfer in just 2 years.
If you are more open minded about the name on your degree, any CSU except SLO and probably SDSU would be happy to have you (assuming you’ve met the a-g requirements).
Good luck