I want to understand what the shell of an atom is exactly. I am not talking about the term shell as it relates to the orbit of electrons. A basketball has an outer shell made of a type of rubber. It is the membrane that separates the air on the inside of the ball from outside air. Back to the atom. Does an atom have such a membrane? If there is such a membrane, does that set up boundaries between atoms? Does the membrane of one atom touch the adjacent atom? Would a basket of atoms look like a basket of tennis balls?

No membrane; each shell consists of clouds of electrons around an atom ( or at least, much more strictly speaking, where the clouds of electrons COULD be around an atom ) with the same principal quantum number:

http://en.wikipedia.org/wiki/Principal_quantum_number

but also see:

http://en.wikipedia.org/wiki/Electron_shell

The outermost shell helps define the outer boundary of an atom but note that this boundary is not well defined but is fuzzy due to probability fields that define the uncertain positions of the electrons.

I assume it is correct to say that a shell ( usually the outer one ) of an atom can “touch” an adjacent atom asp when the electron orbitals form chemical bonds between atoms by hybridizing to form intermolecular bonds.