Correct aspect ratio without re-encoding video file

There is a difference between Sample Aspect Ratio (SAR) and Display Aspect Ratio (DAR). If you want to change the video to display at 4:3, you will either need to change the actual pixels in the image (by scaling the pixels and changing SAR), or by setting a metadata flag that at the container level that tells external media players to stretch the image to your desired DAR.

You will not be able to scale the pixels and change SAR without applying a video filter. If you choose this method, you will be required to transcode the file - since you cannot "stream copy" the video stream while applying a video filter.

To scale the image and change SAR (while transcoding), try:

ffmpeg -i <INPUT_FILE> -vf scale=720:540 -c:v <Video_Codec> <OUTPUT_FILE>

On the other hand, if you just want to change the metadata flag and adjust the DAR, you will be able to stream copy the video. To do this, try:

ffmpeg -i <INPUT_FILE> -aspect 720:540 -c copy [OUTPUT_FILE]

Delgado's answer is correct that MP4Box can do this, but the -par option doesn't work quite as described. With an -out parameter (so as not to disturb your original file):

mp4box source.mp4 -out target.mp4 -par stream-number=width:height

When you use -par stream-number=width:height, you define the pixel aspect ratio – that is, the result of dividing the device aspect ratio by the storage aspect ratio. (Equivalently, you're describing the aspect ratio of a source pixel.) For example, suppose you have a DVD source that's 720×480, and the correct display aspect ratio is 4:3. For this case, you need:

mp4box source.mp4 -out target.mp4 -par 1=8:9

because (4/3) / (720/480) = 8/9.

If the source represents true SD NTSC pixels (in which case only the central 704×480 pixels are supposed to map to a 4×3 screen, with 8 pixels overscan on either side), the correct command would be:

mp4box source.mp4 -out target.mp4 -par 1=10:11

because (4/3) / (704/480) = 10/11 – exactly the reference pixel aspect ratio for standard definition NTSC video.

For the case given in the question, if it's really 4:3, that gives a very odd pixel aspect ratio: (4/3)/(720/416) = 104/135. It's 720 wide, which suggests a DVD source; it's a 25 fps video, suggesting PAL, but the PAR works out to less than 1, suggesting NTSC. It could be 4:5, I suppose (very close to 104:135), but I don't know of anything that produces that pixel aspect ratio; maybe try that first, and then try 3:4 if it still looks a little too stretched horizontally. If you're certain it's exactly 4:3, of course, just use 104:135.

Changing the SAR without reencoding also works with ffmpeg on .mp4 using the h264_metadata as Gyan pointed out here:

ffmpeg -i in.mp4 -c copy -bsf:v "h264_metadata=sample_aspect_ratio=4/3" out.mp4