As the others here have said, creating a (near) vacuum chamber is easy enough, but the weight of such a container is the prohibitive thing.
Atmospheric pressure, which is the weight of the air above pushing down on everything, is about 14.7lbs/square inch. This means that for every square inch of area that an object has, 14.7lbs of pressure are trying to squeeze it together. Let’s say you’ve got a sphere with a radius of 1 inch, inside which you want to create a vacuum. The area of this sphere is 4 * pi * r², which works out to about 12.6 square inches. If the inside of this sphere is completely empty, ie. a vacuum, then the pressure differential between the inside is 14.7lbs/in²:0lbs/in², and with an area of 12.6 square inches, that works out to 185lbs of total pressure spread across the surface.
This seems like quite a lot of pressure, for something around the same size as a chicken egg.
The volume of this sphere is 4/3 * pi * r³, which works out to 4.2in³, which when considering this question is important, because this is the mass of air which would be displaced by the sphere. Things float if the mass of the fluid (or gas) they are displacing (so 4.2in³ in this case) would be greater than the weight of the object in question.
Air at sea level weighs about 1.2kg per cubic metre. Google calculator tells me that this is 4.33527504 × 10–5 pounds per cubic inch, or 0.00004335 lbs per cubic inch. Multiply this by the 4.2in³ for our 1 inch radius sphere, and you get 0.000182 lbs, which would be the maximum weight of your sphere for it to float. That is to say, if it were lighter than this, it would float. The problem is making something so light and strong.
The fleck of gold in this picture http://www.gold-nuggets.org/images/nughnd.jpg apparently weighs 1 gram. Our 1 inch radius sphere would have to weigh 0.082 grams at the most to float. It’s difficult enough making a 1 inch radius sphere of anything which would weigh as little as this, let alone one strong enough that would float. Let’s take gold as an example. It’s not particularly strong, and fairly heavy, but even so, I have the figures to hand…
Gold weighs 19.3 g per cubic centimetre, and we’d need to use a maximum of 0.082 grams of it. That’s about ½35 of a cm³. Spread this over the area of the outside of our sphere (which is 12.6 in², or 81cm²) and you get ½35cm² (0.00426cm³) divided by 81, which is 0.00005259cm, which is the thickness that the walls of your sphere would have to be (made from gold). If you then bear in mind that the average human hair is 0.01, this means the walls of your sphere would be 200 times thinner than a human hair to still be under the weight, while still containing a pressure differential of about half of what’s in an average bicycle tyre.
So yeah, that’s the problem. You try making a sphere with walls 200 times thinner than a human hair, which is also strong enough.