In theory you could do it but it would cost you more than buying one of the mass-market devices. You need to buy the RF and processing chips, display, keypad, etc. Are you going to design and make your own multi-layer circuit board? Design your own antenna? (That's probably the easiest part). I don't think there's any way you can beat buying a handheld unit (they're cheaper than iPods these days).
Do you know how GPS works? You might find it interesting. It's a very sophisticated system that employs spread-spectrum transmission, coding, and fancy math to deduce position from the relative timing of signals from satellites at different positions in the sky. Start with the non-technical description at Wikipedia, then follow the links provided for more information.