The sinkhole is around 4ft deep and 2ft wide and was found in the village of Harrington.
Engineers said the ground around the hole appeared to be "shifting". Services between Whitehaven and Workington were disrupted.
Network Rail, which manages Britain's railway infrastructure, said in a statement: "Engineers are working to repair and make safe the section of line after the mysterious void appeared yesterday in the wake of Storm Eleanor."
Phil James, head of operations for Network Rail's London North Western route, said: "Sinkholes are very unusual on the railway.
"We are trying to work out whether this one been caused by a collapsed culvert or sea erosion or both. Either way, we've got a big hole right by the tracks that we need to fill in.
"We're having to run trains on one rather than two lines along this stretch while we make our repairs. We aim to have things back to normal by Sunday.
"In theory running the railway is simple. But in practice, when you throw storms, wind, waves and sinkholes into the mix, it gets more complicated."
Network Rail has recorded 50 sinkholes on the railway in the past 12 years.
Additional reporting by PA
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