Join an agency, for temporary type roles its more cost effective for them to recruit staff that way. Certain agencies have contracts with a particular local authority etc.
There are probably more temporary agency based public sector admin roles these days than there used to be because they are much less likely to recruit permanent staff due to cut (and declining into the future) budgets.
If you are just doing admin in the public sector though don't expect it to be a bed of roses, I know you hear stuff about how public sector workers get great pay and superb pensions that they can retire on 55, but this is basically doctors, headteachers, senior police officers, etc, public sector admin is a life of low pay and measly pension.
I work in a special needs school as a clerical assistant, i got the job by doing work experience through an NVQ. The school signed up to the training agency to take on volunteers. After a lot of hard work i now work there full time.