Is it legal for my employer to ban work relationships?

In California, the employment relationship is presumed to be at-will which means that you or your employer may terminate the relationship at any time, with or without cause. Nevertheless, employers may not discriminate or harass based on a "protected class" (i.e. race, national origin, age over 40, gender, sexual orientation, disability, religion). Employers also may not terminate an employee if it violates public policy (i.e. terminating an employee who complained about paychecks being late).

Your employer's policy prohibiting intimate relationships between co-workers by itself is not illegal. However, if your employer is enforcing this rule in a discriminatory fashion (i.e. based on a protected class) then it would be illegal.