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The 36-year-old will be relied on to score the goals to fire the Grecians up the league, with the ammunition provided by Walker as well as fellow summer signing Gareth Sheldon.
Goodman last played professional football with Walsall, but left the Bescot Stadium in the summer.
Before the Saddlers he played for Bradford City, WBA, Sunderland, Wolves and Motherwell.
"He is a proven goalscorer at the highest level who has turned down the chance to join several other clubs to come to us," co-chairman JOhn russell told the Express & Echo.