Hi there,
Check to make sure the transmission lever is in Park or neutral. Actually, try jostling the shift lever back and forth in case the park/neutral switch is dirty. If that doesn't do it, open a door and see what happens to the dome light when you turn the key - if it goes out, check the battery connections - occasionally a battery connection will have enough resistance that it will fail to turn the starter, but will still allow enough current to flow that things like the dome light, radio, etc will still work.
If that doesn't do it, the starter is likely the culprit.

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Richard Wooldridge
'01 ML320
'82 300D 4.3L V6/T700R4 conversion
'82 380SL, '86 560SL engine/trans. installed
'79 450SL, digital servo update
'75 280C