Re: If something has mass, does that mean it also takes up space?

Our notion of matter and reality are greatly different from
those of classical physics, i.e. the paradigm that is taught in
schools. I've tried not to use of the word "matter" since classically
this excludes light. The division between classical views of light and
matter become somewhat blurred in Quantum Theory (QT). Light and Matter
are still very different in QT, but QT states that everything exists as
wave-particles and since your question is one of existence i feel
justified.

Question (1) If something has mass must it also take up space?

 Physicists describe the universe with only a handful of
theories. The most fundamental of these, when we are talking of the very
small, is Quantum Theory (QT). As such it seems like a good theory to
use to answer your first question. In QT (or more Specifically in the
Standard Model of Particle physics) all fundamental particles like the
electron or quark are considered to be point like particles. That is in
all calculations it is assumed that all fundamental particles have zero
size or take up zero space.  If we were being hasty we may say that the
answer to question 1 is that it is possible for an object to have zero
size and yet have mass; the electron being an example. This however may
not the case.
  Our most accurate measurement of the size of an electron finds no
structure but this is very different from stating that it is a point
particle. What  if we make a better measurement?
    In fact in an attempt to understand nature on a deeper level String
Theory is being developed. In this theory particles can crudely be
thought of as oscillating strings that are admittedly very small, 10^-35
m, but definitely non zero.

As an aside for the more clued up;

    There is a hint of a fundamental limit to how small anything can be
said to be  in QT. There is a distance known as the Planck length which
is believed to be the scale at which Quantum Gravity becomes dominant.
Associated with the Planck length is the Planck time, the time taken for
light (or rather information) to traverse the planck length. The nature
of such things is the answer to a different question so I hope you will
forgive me for simply using them.
    What is important is that there exists a quantity called the Planck
length and a quantity called the Planck time collectively they are known as
the Planck scale. At the Planck scale space and time lose all meaning. It
can be said that NO interval of time exists below the Planck time and NO
interval of space below the Planck length. Our best guess is that at this
level space-time becomes like an ever changing foam and new theories take
over from QT; possibly String theory. Essentially space-time itself does
not exist below the Planck scale. If space does not exist on length scales
less that the Planck length then how can we claim that something is
smaller? Of course we cannot. Extension below the Planck scale becomes as
meaningless as time before the beginning of time or south of the south
pole. On this level question (1) has no meaning.

    In summary to answer (1) I hopefully have shown that it is believed
that all objects take up space irrespective of mass. Current theories
may assume point like particles but this is expected to be an
approximation, albeit a fantastically good one! This either appears to
be true or at the very least the notion of extension becomes questionable.
Either way, it can be said that no object can be observed as having zero
size whatever the mass may be.

Question (2) If something takes up space must it also have mass?

No
This ones a lot easier to answer because we have examples. For instance
the Photon has zero mass. Part of my answer to question 1 was
essentially that no object could be observed with zero size.
Consequently the photon can not be observed to have zero size even though
QT currently treats it as such. The photon is known to be massless and
hence is an example of an object that has extension yet no mass.


Incidentally, as a footnote, Rene Descartes defined matter as;

    "Extension in length, breadth, and thickness constitutes the nature
of corporeal substance"

he explained: "Everything else that can be ascribed to a body
presupposes extension". That is mass, charge, color, texture, motion, or
any other property are incidental . Substance can exist without all of
these if, and only if it takes up space. It is interesting to find that
a seventeenth century philosopher could answer your question in one line
where as myself and 300 years of Physics took a page!