B767-300 FLT CREW ATTEMPTS TO FLY AN APCH FOR WHICH THEIR ACFT WAS NOT EQUIPPED.

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ATIS WAS RECEIVED 30 MINS OUT. BTWN THEN AND OUR APCH; THE ATIS CHANGED FROM LNDG S TO LNDG N. VERY BUSY ON APCH WITH MULTIPLE INBOUNDS. I KNEW WE WERE NOT GPS EQUIPPED; BUT WITH THIS WX; IF WE COULD GET BELOW THE 4500 FT LEVEL; WE COULD SEE THE ARPT. WE HAD BEEN IN VMC CONDITIONS FOR THE LAST HR; AND COULD SEE THE GND AND LIGHTS THE ENTIRE TIME. I TALKED WITH MY FO AND RELIEF PLT ABOUT USING OUR FMC LNAV (GPS UPDATED) TO GET US LINED UP ON FINAL AND IF WE COULD NOT SEE THE GND; WE WOULD GO AROUND. THE FO MADE A STATEMENT ON DOWNWIND THAT SHOULD HAVE SET OFF AN ALARM; 'CAN WE DO THIS?' SINCE WE WERE IN VMC CONDITIONS; I TOLD HIM IT WOULD BE OK. WE WOULD NOT DSND UNLESS WE SAW THE RWY. HE AND THE RELIEF PLT SAID OK. THAT SHOULD HAVE BEEN MY CLUE. WE WERE CLRED TO 3000 FT STILL VMC; LINED UP ON FINAL; CLRED FOR THE GPS RWY 35R APCH. I DSNDED TO 2400 FT (INITIAL APCH ALT) AND WE WENT IMC. WE WERE 1.3 MI OUTSIDE OF THE FINAL APCH FIX; LEVEL. I ASKED THE FO AND RELIEF PLT IF THEY FELT IT WOULD WORK. THEY AGREED IT WOULD NOT AND WE BEGAN THE GAR. GOT IN LINE FOR ILS RWY 35L (WHERE WE SHOULD HAVE BEEN IN THE FIRST PLACE). I THINK MY THOUGHT PROCESS WAS; I COULD SEE THE GND; THE FMC WAS GPS UPDATED; SO I FELT LIKE I WAS WHERE I SHOULD BE. I WAS FAMILIAR WITH THE AREA AND IT WOULD WORK. IN ADDITION; THE LAST ATIS JUST PRIOR TO THE APCH HAD A BROKEN LAYER AT 1100 FT. AGAIN MY BRAIN THOUGHT SINCE I COULD SEE THE GND IN EVERY DIRECTION AROUND THE ARPT (EXCEPT OVER THE ARPT) THAT WE WOULD BE ABLE TO MAKE GND CONTACT (RWY IN SIGHT) OUTSIDE THE FAF AND LAND. I SHOULD HAVE LISTENED TO MY FO. HE GAVE ME THE CLUE; I DIDN'T LISTEN. I WAS OVER-CONFIDENT IN MY ACFT ABILITIES AND BELIEVED THE WX WAS NOT THAT BAD. IT WAS; BUT ONLY OVER THE ARPT. I SHOULD NOT HAVE ACCEPTED THE APCH. AFTER WE MISSED; THE CTLR ASKED IF ANYONE ELSE ON THIS SIDE OF THE ARPT COULD NOT ACCEPT THE GPS APCH. 3 OTHER FLTS SAID THEY COULD NOT. I WONDER WHAT THEY WERE THINKING ABOUT?