I've created a rudimentary Gantt chart.

Days are listed along row 2 in the form DD-MMM-YY
and Task are listed down column A.

I've marked with an "x" the cells which represent the
days activity is happening as part of each task.

I want to create a Start date column and an end date column.

Here's one way, with Gantt data in row 3.

Start date:
=INDEX($F$2:$BN$2,1,MATCH("x",F3:BN3,0))

End date:
=INDEX($F$2:$BN$2,1,MATCH("x",F3:BN3,0)+COUNTIF(F3 :BN3,"x")-1)

Copy down for subsequent rows.

Hope this helps getting started.