B737 CREW DEPART GTR AFTER THE VOID TIME ISSUED WITH IFR CLRNC.

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TOOK OFF AFTER VOID TIME. VERY; VERY; VERY RARELY OPERATE INTO UNCTLED ARPT. CAN ONLY REMEMBER ONCE TAKING OFF FROM UNCTLED ARPT. CONTACTED NEARBY APCH CTL FOR CLRNC; VERY RAPID FIRE RESPONSE INCLUDED A VOID TIME OF XX32? WE WERE; AT THAT POINT; STOPPED ON RAMP WITH ENGS OFF. FAIRLY SHORT TIME AVAILABLE; BUT NO OTHER TFC ON ARPT. QUICKLY STARTED AND TAXIED OUT; BUT TAXI WAS SLOW DUE TO VERY NARROW TXWY. I WAS SCRAMBLING TO COMPLETE ALL CHKLISTS AND LOST TRACK OF TIME. I DID MANAGE TO MAKE ALL REQUIRED CTAF RADIO CALLS AND DO ALL CHKLISTS. I WAS PF FOR TKOF. AFTER TKOF; CAPT CALLED APCH AND WAS SCOLDED FOR TAKING OFF 4 MINS LATE. NO TFC CONFLICT. GTR IS A VERY MARGINAL ARPT (TXWYS; MOSTLY) FOR A B737. COMBINED CHALLENGE OF UNFAMILIAR; MARGINAL ARPT; UNFAMILIAR NO-TWR OP AND I JUST LOST TRACK OF THE TIME. IN OVER 13 YRS OF AIRLINE FLYING I HAVE NEVER DONE JUST SUCH AN OP. AT THE LEAST; TWR ALWAYS CHKED AND ENFORCED ANY VOID TIME. I GUESS I AM JUST USED TO HAVING TWR LOOK AFTER US ON THIS ISSUE; AND WE ARE FORTUNATE NOT TO HAVE CAUSED A TFC CONFLICT. MIGHT HAVE HELPED IF APCH HAD ASSIGNED A MORE REALISTIC OFF TIME FOR A LARGE (RELATIVELY) JET OP. WE WERE ACTUALLY FAIRLY EXPEDITIOUS IN STARTING AND TAXIING OUT; SO I THINK THE TIME ALLOTTED MUST HAVE BEEN FAIRLY SHORT.