Are you looking for a sound level meter - one that gives and instantaneous indication - or a dosimeter that records sound levels over a predefined period so that some understanding of the overall "dose" of noise is indicated?
For a simple noise level meter a microphone and sensitive amplifier (lots of kits from places like Vellman) might work. Output of amplifier could be indicated with voltmeter (with appropriate load resistor). If you had a mic and a stereo with a mic or other suitable input you might try it to see if you can actually pick up whatever it is that you want to measure. It won't be accurate, linear, etc but might be better than nothing.
If my understanding of a dosimeter is correct then construction of a dosimeter would be quite a challenge. A poor man's version might be fashioned from the mic/amp described above then tied to a data aquisition card ( I got mine for $24, slow but it works). You'd have to decide how you process the data you collect.
Things I'd worry about with this kind of approach is frequency response and changes in amplifier output with time - as the mic/amp warms or cools, power supply voltage changes, etc.