Is there a size of rectangle that retains its ratio when it's folded in half? A hypothetical (and maybe practical) question has been nagging at me.

If you had a piece of paper with dimensions 4 and 3 (4:3), folding it in half along the long side (once) would result in 2 inches and 3 inches (2:3), which wouldn't retain its ratio. For example, here is a piece of paper that doesn't retain its ratio when folded:


Is retaining the ratio technically possible? If so, what is the side length and ratio that fulfills this requirement? Any help would be appreciated.

Update:

I added "once" because I got an answer saying that any recectangle would work, as any rectangle folded twice has the original ratio. Nice answer, but not quite what I was looking for. As for the other answers, I got 3x as much information as I needed! Thanks!
The $1:\sqrt{2}$ ratio ensures exactly that. That is the idea behind the ISO 216 standard for paper sizes, which was adopted from the German DIN 476 standard.

Its most common usage is the A series which especially in Europe is a collection of very common paper sizes. The base size, A0, has an area of a square meter, and every next smaller paper size is constructed by folding it in half.
Well, if you have paper that has a ratio equivalent to the square root of two with the length divided by the width, if you fold it in half hamburger style (from width to width), you will get another ratio that's equivalent to the square root of two of the same property, likewise if you fold it more in half that way.

How It Works:

$$\sqrt{2}/1=\sqrt{2}$$$$\sqrt{2}/2=1/\sqrt{2}=0.7071...$$

That's half the square root of two, and when one is divided by it, we get the square root of two again: $1/0.7071...=\sqrt{2}$ and it just keeps going on and on from number to number being divided by the square root of two to get that result.  Just divide the dividend by the quotient in those things.