Let's assume that we build a giant steel hull in a shape of cube with open top (2km long edge) and lift it to the top of stratosphere and then pump air out of it. Would it float on the outer layer of stratosphere like a ship floats on the surface of water?
You need to consider the pressure on the cube. The cube would have a certain initial average density (mass of the steel divided by the volume of the cube). The cube would sink to at least a level of its average density. Then the pressure would begin to crush the cube, decreasing the volume and increasing the density. The cube would progressively be crushed and fall to Earth, like this.
Also, if it is an open top cube, it would start filling with low density gas, increasing the average density of the cube.
Nice question! The answer is yes, it is certainly possible. There is no need to start at the top of the atmosphere though. The air around the earth has a density (and pressure) gradient that increases as you get closer to the surface, see the figure below. If you have an object whose average density is less than the atmosphere at the ground, then it will rise to a level where its density is equal to that of the atmosphere.
This is the same reason that, for example, oil and vinegar separate over time in a jar. It is also the reason that hot air balloons and blimps are able to fly.