Extract sound file from electronic answering machine
General procedure starts with a schematic. If you can't find one (likely) then you will have to 'reverse-engineer' it by tracing the wiring inside the device.

Necessary equipment equipment includes a screwdriver, continuity tester or multimeter, pencil and paper, and a good pair of eyes. Some basic electronics knowledge will also help.

Once you have a schematic you should be able to determine where the audio signal is produced and amplified, then you can try probing different points to find the best signal. 

Or just take the signal directly from the speaker terminals - on the assumption that the electronic signal is relatively high quality and most of the degradation occurs inside the speaker itself. Even in this case a schematic is still useful, as it may help to determine what kind of interface is required.