Did Gustave Eiffel build the Sydney Harbour Bridge?

No. Gustav Eiffel had no input into the building of the Sydney Harbour Bridge.

Engineer JCC Bradfield prepared the general design of the bridge. Construction was undertaken by the firm Dorman Long and Co Ltd, Middlesbrough, Teesside. Sir Ralph Freeman, of Freeman Fox & Partners, was the designer of the Sydney Harbour Bridge. Another civil engineer involved in the Sydney Harbour Bridge project was JF Pain, who was awarded the Manby Premium for a paper on the Sydney Harbour Bridge. He led a group of about forty engineers working for Freeman on Sydney Harbour bridge.