Why can't you flatten a sphere? It's a well-known fact that you can't flatten a sphere without tearing or deforming it.  How can I explain why this is so to a 10 year old?
As soon as an explanation starts using terms like "Gaussian curvature", it's going too far for the audience in question.
A great explanation would work just as well for a hemisphere as for a sphere, as both have positive curvature.
 A: Stand with your right arm pointing forwards and the thumb pointing up.
Swing your arm upwards till it is vertical, then to the right until it is again horizontal and swing it back so it is pointing forwards.
Now your thumb is pointing to the right, but relative to the surface of the sphere on which you have been moving your hand, these moves all kept the orientation of the thumb.
Now consider what would happen if we flattened the sphere without tearing it, and tried to do the same sequence of moves there.
A: I've got another idea akin to the orange peel experiement mentioned by Ovi. At its heart, it is talking about curvature.
Look up with the student, online or otherwise, what the distance R from the geographic north pole to the equator is. Convince your student that the equater is "a circle centered around the north pole," since all the points on the equator are equidistant from the north pole.
Now look up the length of the equator, and compute for them the circumference of a circle radius R in a plane. Point out that although the radii of the circles are the same, the circumference isn't the same. Argue that the circumferences should be the same, if you are going to flatten without stretching things.
To provide a constrasting example, you can draw a flat circle on a piece of paper, and then go between rolling the paper into a cylinder and flatting it out to show that that circle can be flattened, and that the radius and circumference are indeed unchanged.
A: How about using a counter example:  Take a flat sheet of paper and form it into a cylinder and a cone and demonstrate that you can unroll it to make it flat.  Now if you do the same with a ball of some sort, it should be obvious to the child that it cannot be unrolled in the same way.  This will help them understand the difference between intrinsic and extrinsic properties.  Although the cylinder and cone are curved, their curvature is different.  You can then explain that on a cylinder and cone you can always draw a line that stays straight when the paper is rolled up.  In contrast, on a ball the lines are always curved.
A: Experimentation is fun!
Give him a globe, some string, a ruler, a compass, some paper, and a pencil.  Then have him try to map 4 different cities from the globe onto the paper, keeping ALL the distances between them the same as on the globe.  
A: Show him a triangle on the sphere with 3 right angles. He knows that it is impossible in a plane right ?
A: Which kind of flattening do you mean? If it's about bounded flat surfaces, then imagine a fly captured inside Git's balloon. Would it be possible to flatten the balloon without deforming it, the fly would do this an escape - but it can't. However, it does not help in case you allow the flat surface to be unbounded
A: You can discuss curvature with a child. You do it by taking a flat piece of paper, the previously mentioned orange peel and a mustard leaf. From each you cut an annulus and make a transverse slit in it. You show that the flat piece of paper's annulus can be laid flat with no adjustment. You see that the orange peel's annulus opens up when you lay it flat. You explain that that angle is the curvature. The mustard leaf will be more in interesting. When you lay it flat you will find there is excess annulus. You can then explain negative curvature.
A: One simple way is to think of an orange peel. Have the 10-year old peel an orange nicely and tell him to try to flatten the peel pieces completely without breaking or streching any part of the peel.
A: For a 10 years old :
Take a  fix sphere (half of it,  a quarter)  and a sheet of paper. 
Then ask the child to perfectly "flatten[1]" the paper on the sphere => it is impossible. 
Of course because, the sphere has a curvature and the sheet does not have.so they are not locally isometric
That is actually the theorem (for any portion of the sphere and arbitrary small sheet), because the sheets of paper can only be deformed isometrically unless you start tearing it of. 
you can make them try the same with a cylinder to see the difference. 
[1]: It was the  translation I found for "plaquer" hope it is not misleading.
A: I take it that you will be explaining this to a group of kids. Why don't you try several approaches? Some will probably appreciate the orange peel explanation simply because it gives them something to do. Others may appreciate the following. Mathematically it is based on the fact that the circumference of a circle on a spherical surface is shorter than the circumference of a circle (with same radius) on a flat plane (see also rschwieb's suggestion of calculating the ratio of  the length of the equator to the distance to a pole). So changing the radius by a fixed amount will not result in a fixed change in the circumferemce.
I suspect some of the kids in your audience will have seen just that while crocheting. If you are crocheting a circular piece (like this), then you have to be careful to add just the right number of links per round. Add too few, and the end result will have a positive gaussian curvature (a bag-like shape) - add too many and you get a warped surface of a pseudosphere (was still good enough, when yours truly handed such a potholder to his dear Grandma 40+ years ago).
Undoubtedly you will have shown how to flatten a cylindrical surface prior to this (also suggested by rschwieb). 
Also make them gift wrap both a cylindrical tube and an inflated basketball.
A: (This is a stab in the dark, but let me know if you try it!)
I'm tempted to say "Read Flatland by Edwin Abbott", but that will take some extra explaining.
Have them draw an outline of their hand on a piece of paper, cut the drawing out of the paper.
Now, ask them to compare the the width of their hand to the "width" in the drawing. (the axis from their thumb to their pinky). Then have them compare the "height" of their hand to the one in the drawing (wrist to the tip of their middle finger). 
Hopefully, they will notice that they are about the same.
Now, what about the "thickness" of their hand? Is it the same as the paper? Probably (hopefully) not.
How would we make it so that the "thickness" was the same?
Well, one way would be to use a rolling pin and squish their hand as flat as the paper.
Spheres work the same way, when you look at them from one direction, they have a width and a height, but when you look at them from another angle, they have a thickness, like a person's hand.
You might consider repeating by outlining the hand clenched in a fist, or get a ball of dough and use a rolling pin. Look at the mess it will make, the dough must go somewhere when you reduce the thickness (a dryer dough that will crack will demonstrate this better).
A: Get a basketball and cut it open (make sure it stays as one piece) and try to flatten it, it will not flatten ... there will always be some part of the basketball that is bulging.
A: Imagine that you have a sphere and a plane (a ball and a floor). Draw an equilateral triangle on both of them (same side length for both).
The area of the triangle on the sphere is greater (this is apparent if the triangle is big enough). If you wanted to flatten out the sphere, you'd need to make the bigger triangle fit inside the smaller one, which you can't do without squashing (projecting).
Alternatively, think about the film of soap when you blow soap bubbles (a plane is deformed). The ring on the outside never grows, but the surface area of the film evidently does. The message is that "bulges" have greater surface area than proper, law-abiding euclidean planes. Again, to flatten out a "bulging" (curved) shape, you'd need to fit a big area into a little one, which you can't do. (In this spirit, of course, a sphere is just a big bulge.)
(these do require him to intuitively grasp that his "flattening out" requires isometry, and that any kind of "squashing" breaks the isometry.)
If he wants to know why the triangle on the sphere (or bulge) is bigger, first make a note about the shortest distance between two points being a straight line. You can go on to observe that all lines on a sphere surface, when viewed in 3D space, are actually not straight, while those on a plane are. So because all lines on a sphere are curved, they are also "longer than they need to be", and since areas are products of lengths, it follows that the areas on a sphere are also larger.
A: Being a cartographer, it becomes second nature to think that you simply can't do that without introducing distortion of any kind, yet it's always a struggle to explain.
B.J.S. Cahill's 1913 rubber-ball globe, might be even better than peeling an orange. Sadly they are currently not made or sold anywhere to my knowledge; desperately hoping for a kickstarter.
I think the most hands on explanation/visualization could be the act of trying to flatten a hemisphere-shaped object like:


*

*a beanie

*an elastic rubber mixing bowl or as Pykler said half a basketball

*a more rigid mesh like a geodesic dome†


You might notice, that you can't change the form of the dome without breaking links‡.
With the beanie or elastic cup it's notable that the mesh/rubber changed its form, and any pattern drawn on it has been distorted, after it has been pressed into a flatter state. 
Simply put, there is no projection without distortion, since by definition all points on a sphere have the same distance to its center, the closest you get to this in flat space is a circle yet that's about it.
† a nice little hands on project by itself
‡ plus this might explain how Buckminster Fuller came up with his Dymaxion map
A: You can, just imagine a perfect sphere, being put into a piece of paper that breaks away as it enters, then take a picture of it. Perfect sphere, the reason you can't physically flatten a sphere, is because all of that third dimension of space has to go SOMEWHERE, so it's either going to fall apart, and split the third dimension perfectly (EXTREMELY unlikely) or, it'll stay together, and become deformed.
A perfect sphere is merely a circle anyway, it's just a filled in surface of the spheres diameter.
This is a complete shot in the dark, and I hope my answer helps, but it's obvious so it probably won't :/
