What was the standard size of home videos in 1991 and how can I stretch it to fit modern day screen - without losing quality?

Really, you cannot without using a proper Editing Program.
Expanding the image, which will cut off either or both the top and/or bottom of the images.
Although, yes a minor sideways expansion is possible.

Most people tend to get too much "nothing" of value in the upper part of the captured images.  Note in your displayed image that the Nirvana (over the heads) of the crowd could be cut.

When I watch movies on my Sony Bravia 55 inch LCD TV they show in 4:3 ratio because if I use widescreen the movie will be cut off.  

the VHS standard was approx. 640 x 480 pixels interlaced.
it was basically 4:3 (4x3) sizing.you cannot stretch video and maintain its quality. it doesn't work that way.
the only way to capture video and maintain quality after resizing is the hard way: frame by frame.
the best way that is still faster than frame-by-frame, is to use a high end SVHS VCR with TBC (Time-Based Correction), and then capture using a decent capture card to the computer at a very high bitrate.
if you try to make it HD from VHS you will fail. best you can do without lots of time wasted is DVD. then just use an upconvert DVD player to "stretch" project it to your big screen...

Standard TV resolution was 4:3 at 480i for NTSC or 576i for PAL.
The black bars (pillarboxing) are there because that gives you the truest image quality without stretching or losing any content. If you stretch the image to fill a modern 16:9 screen, then everything looks wider than it should. Some TVs will have a "zoom" mode that focuses on the middle of the image and cuts off the top and bottom so that what's left fills the whole screen when upscaled.
The other thing you can do with video editing is to trim the top and/or bottom of the picture yourself so what's left is 16:9, but while taking notice of what gets cut so that you don't lose anything important. You don't lose quality, but you do lose that content that gets cut.

I'm too old but as far as I can remember, wide screen TVs weren't that popular yet that time so I'm guessing videos would have an aspect ratio of 4:3. Those black bars on the sides wouldn't have appeared then. All TVs today are wide so they have a feature that can adjust aspect ratio on the fly. You can choose (among other options) wide (which puts black bars above and below the picture which is complete) or stretch (which fills the screen but portions of the picture are trimmed).