The 35-year-old former Leicester midfielder has started all but two of the Grecians' League Two matches this season and been a vital cog in the side that sits just two points off the automatic promotion places.
Manager Paul Tisdale, a former team-mate of Oakley's at Southampton, is already without fellow midfielder Liam Sercombe (broken ankle) and Irish striker John O'Flynn (hamstring).
"He will be out for a while," Tisdale told the Express & Echo. "He will be out for a few weeks at least and that will be a big loss for us as it is another midfield player. Most teams have gone through January and tried to strengthen their squad and it seems we have weakened really.
"We had the signing of Mark Molesley, but with Liam Sercombe, John O'Flynn and now Matt all dropping out it has proved to be quite a hard-hitting month for us in terms of our squad."
Tisdale added that financial restrictions at St James Park mean he does not have the option of signing a replacment for Oakley.
"There is no point grumbling about it," he said. "We've got good enough players to play, it is just the depth of the squad that has been weakened."