Can anyone tell me what physical form a quantum computer is likely to
take?  What will the core consist of?  How large will it be?  How
stable?  (Will it need a constant power source to retain memory?)

It may look as any other computer. Computer  has many components, CPU,
mass storage, peripherals. Most of these will be same as they are
today.
http://computer.howstuffworks.com/pc1.htm

Quantum computing, so far, is limited to the CPU, 
which has volatile memory only. 
http://www.webopedia.com/TERM/N/non_volatile_memory.html
Today it is a technology, not a product
http://computer.howstuffworks.com/quantum-computer3.htm

However, we can guess how such computer will look, as at least initially,
since first machines will be supercomputers and will need very low
temperature to operate.
First supercomputers (cray1 cray2, ..) already has such cooling system
news.bbc.co.uk/.../ newsid_1484000/1484268.stm
http://accl.grc.nasa.gov/archives/cray3-ymp_c-90-08063.jpg

Todays supercoputers did abandon this branch of evolution and use clusters
of ordinary (room temperature) CPUs
http://www.newsfactor.com/perl/story/19391.html

 Quantum computers will again need the large cooling system and so,
chances are that first such machines will look like cray 2 in more ways then one.
 Here is more on history, looks  and design of that particuar  machine.
  http://www.spikynorman.dsl.pipex.com/CrayWWWStuff/index.html

From a little I know about quantum computing, the quantum computer
ideally would have to be extermely small basically the size of an atom
or a molecule.
But it may not be possible to create such a computer because of the
inherent nature of molecules.But some scientists have proposed a
different approach to solve the problem by theoretically proving thats
its possible to take a cup of liquid and assume each molecule to be a
quantum computer. By building in such massive redundancy in computing
since erratic behavior of some molecules will not affect the
statistical behavior of all the molecules in the cup, the quantum
computing can still be acheived.