I had the very same problem with my '88 300E about two months ago. After several tests, it turned out to be the fuel pump relay. The relay controls other items aside from the fuel pump. It is relatively simple to change and it ended up costing me $229 with labor and diagnostics.
'86 Alfa Romeo Spider
'84 Harley Davidson FLH"X"
'70 FJ40 Land Cruiser (basket case)