I think the ultimate souvenir would be not a moon rock or a martian, but space "nothing", but I wonder if this is physically possible... Can an astronaut go out to space with an open container, close and seal it and bring it back to Earth? Is there a container that could withstand the pressure of Earth's atmosphere upon reentry? If this is possible, what does my container of space "nothing" truly consist of?
Space is not quite as empty as you might think, and your average jam jar is easily strong enough to hold atmospheric pressure against an internal vacuum. More to the point, there wouldn't be any need to go into space to create a jar full of nothing. Current technology can easily create a vacuum that would compare with that of "outer space".
As for what your container will contain, it depends on your point of view. In everyday terms, there is nothing, except perhaps a few stray particles bouncing around. If you agree with the relevant underlying theory, your container is full of quantum foam. But then, the quantum foam is everywhere, and the container just happens to be empty of anything else.
If you open the container in space, the gas contained in the container will flow into space. At first there will be many atoms flowing out, the number will decrease with time very fast. It would take a pretty long time until the number of atoms per volume will be the same inside and outside the container. If there is only a small hole in the container, the probability for an atom to fly in the exact direction of the hole is small, atoms missing the hole will bounce back from the container inner surface until they got the right direction.
But the container will outgas depending on its material. Even a material with very little outgas will release some atoms into the container.
Due to outgasing the vacuum inside the container will be not as good as in space. The vacuum of space is not perfect anyway. If you close the container, outgasing will continue and the vacuum in the container continues getting worse. When the container is brought back to the surface of Earth, there will be some leakage through the seal of the container's cover. So the number of atoms flighing in the container will increase over time.
So it is impossible to bring back "nothing". There are more atoms in space than "nothing" and the container inner surface and the leakage of the seal will add more and more atoms to "nothing".
This is a "Seinfeld" topic. Seinfeld was famously "the tv show about nothing"
It is easy to build a simple container to faithfully capture and hold "space". The pressure differential between deep space and earth sea level is trivial in comparison to pressures experienced by manned deep sea craft. The problem comes about when trying to keep the "space" pure over time. As mentioned by others, atomic migration from the container material, seal material and thru the container seal will ruin the sample. A high quality welded metal hermetic seal or a fused glass seal can go a long way to providing extended storage time but the act of welding or fusing would contaminate the sample.
One possible option would be to use a double seal. The initial seal would be a soft metal compression seal using a soft, low activity metal like pure gold. This initial seal would be used only to keep the contaminants created by the permanet weld or fusion from getting to sample. A simple glass jar and glass lid would be easy to fabricate.
For the EKV (Exoatmospheric Kill Vehicle) program we used Indium rather than gold for a compression seal on our part of the project.