Is there a globe with the exact shape of the Earth?

To scale? It's about 1000-to-1 ratio from Earth's diameter to the tallest mountains. A 1 m globe would have 1 mm tall mountains.

I saw a video on YT that said basically if you scale the Earth down to something you can place on a table, it would still feel smooth to the touch. The globe would have to be some really huge number before you could detect features by touch.

Unfortunately I don't remember what channel that was. But they did a lot of comparisons to show why. You can pretty much get that idea just looking out a jet window, since the Earth already looks pretty smooth even at that close range.

While everyone else had mentioned that Mt Everest would barely be a blip when making a round globe, I want to bring up another way the Earth is non-uniform as a sphere: The elipsoid shape, in that Earth is wider around the equator than it is on a north-south axis[1]. That's attributed to the rotation of the earth, and it's about 70,000 feet - Or about 11 miles. Out of a diameter of 8000 miles, I don't think you'd be able to spot the difference in the same 1 meter sphere that people are holding as an example.

It'd make a very cool art project, though.

https://oceanservice.noaa.gov/facts/earth-round.html

Are you looking at the specific mountain range shapes too, or simply the overall gist of the ball-like part?

I wonder if it could be 3D printed with reasonable precision to suit your needs. I'd look at shapeways, and if you can't find it maybe somebody can 3D model it for you first.

https://www.shapeways.com/marketplace?tag=earth

How do you define "exact"? What are you measuring?