First remove the audio track *

⇒ ffmpeg -i input_file.mpg -vcodec copy -an copy output_file-audio.mpg

Then turn our multiplexed mpeg-1/2 file into an H.264 encoded mp4

⇒ ffmpeg -i input_file.mpg -strict experimental -vcodec libx264 -an -ab 160000 -ac 2 -preset slow -crf 22 output_file.mp4

This also seems to work if you want to keep the audio

ffmpeg -i input_file.mpg -vcodec h264 -acodec aac -strict -2 output_file.mp4

Bonus code for converting a folder full of mpg's

for x in *.MPG; do
    ffmpeg -i $x -strict experimental -f mp4 \
           -vcodec libx264 -acodec aac \
           -ab 160000 -ac 2 -preset slow \
           -crf 22 ${x/.MPG/.mp4};
done

* Not necessary, just happened to be requested