In a sense, you can. Though that might not mean quite what you think it does. It is a bit like standing in the beam of a flashlight. The video you linked shows this happening - Drove Through the End of a Double Rainbow

Following a few more links, this answer to Rainbows and Clouds is a good illustration of what is going on. Think of the sun as a flashlight, and raindrops as a sort of combined mirror and prism. You see a beam of reflected sunlight that has been broken up into colors. You don't see an object. You see a beam of light coming at you from a narrow range of directions. As long as you are standing in the beam and looking upstream into it, you see a rainbow.

As you move around, the upstream direction doesn't change. It is determined by the direction of the unreflected sunlight. But the scenery in that direction changes. Mentally you convert the moving scenery you see into the stationary world. Since the rainbow moves against that stationary background we see the rainbow as moving.

We are not used to seeing light coming from a direction. We think in terms of light coming from an object. We try to use parallax to figure out where the object is, and it doesn't work. It makes rainbows a little confusing. There really is no object. So in that sense, you can't stand inside the object.

Sometimes the rain is a distant patch. You are only in the beam when the direction from you to the patch is just right. So you only see it briefly as you drive past.

But once in a while, the sun shines into a patch of rain falling right on you. In that case, you are standing in the middle of the mirror/prism. The reflection happens right in front of you, all around you.

You could see this happening in the video. A truck passed a few feet away, and reflections were happening between the camera and the truck. It is extra intense because it is happening in the spray kicked up by the truck's wheels. The spray lands on the windshield, so the reflection is happening that close. I am willing to consider that being in the rainbow.

In a sense, you can. Though that might not mean quite what you think it does. It is a bit like standing in the beam of a flashlight. The video you linked shows this happening - Drove Through the End of a Double Rainbow

Following a few more links, this answer to Rainbows and Clouds is a good illustration of what is going on. Think of the sun as a flashlight, and raindrops as a sort of combined mirror and prism. You see a beam of reflected sunlight that has been broken up into colors. You don't see an object. You see a beam of light coming at you from a narrow range of directions. As long as you are standing in the beam and looking upstream into it, you see a rainbow.

As you move around, the upstream direction doesn't change. It is determined by the direction of the unreflected sunlight. But the scenery in that direction changes. Mentally you convert the moving scenery you see into the stationary world. Since the rainbow moves against that stationary background we see the rainbow as moving.

We are not used to seeing light coming from a direction. We think in terms of light coming from an object. We try to use parallax to figure out where the object is, and it doesn't work. It makes rainbows a little confusing. There really is no object. So in that sense, you can't stand inside the object.

Sometimes the rain is a distant patch. You are only in the beam when the direction from you to the patch is just right. So you only see it briefly as you drive past.

But once in a while, the sun shines into a patch of rain falling right on you. In that case, you are standing in the middle of the mirror/prism. The reflection happens right in front of you, all around you.

You could see this happening in the video. A truck passed a few feet away, and reflections were happening between the camera and the truck. It is extra intense because it is happening in the spray kicked up by the truck's wheels. The spray lands on the windshield, so the reflection is happening that close. I am willing to consider that being in the rainbow.

Addition from Graham's answer (+1 to him)

Sometimes you get intense colored beams of light from prism shaped glass in a building. That is very like a rainbow. The beams can be intense enough to light you up with colored light.

A rainbow is usually not bright enough to do that enough for you to see. But it does shine colored light on you. If you can see a rainbow, you are lit up with the rainbow's colors.