The vast majority of jobs don't require a CRB check and (since your conviction is 'spent', for most purposes, under the provisions of the Rehabilitation of Offenders Act) you can lawfully answer 'No' to any questions about criminal convictions (except for employment applications in 'exempted' areas, such as working with or alongside children or vulnerable adults). So I'm surprised that you've been turned down for jobs, since you don't need to tell most potential employers about your conviction and most employers should have no access to a CRB check.
There's no way you could now appeal against conviction. You should simply concentrate on applying for employment in those fields which don't require a CRB check (which still leaves the vast majority of jobs open to you).
Chris