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Why not just call into your voicemail from a land line, listen to the messages that you want to save, and use a digital recorder to record that call?
Failing that, assuming that you have a line-in input on your computer, just get a 3.5mm male/male audio cable (you might also need to get a 2.5mm adapter, depending on your phone), connect the phone to the line-in on the computer and use an audio editor like Audacity to record (digitally) and edit the voice mails for posterity. Then just kick back and enjoy listening to your old voice mails again and again.
The third, easiest and lowest quality option would be to just use a decent external mic with your computer and record the the messages while you play them back though the speakerphone.
As for video, I don't even know how you would transfer those files from a feature phone without either a USB connection or an SD card, except for emailing them to yourself as attachments.