Good Day All, 
My question is regarding accessing a computer (with or without firewalls).
Most people who subscribe to cable or DSL internet service leave their
PC's on 24 hours a day ("always on").  This leads to possible
intrusions by those seeking systems to hack.
I have heard people say that even thought a computer is SHUT DOWN and
powered off (no power except CMOS battery, but STILL plugged into the
power outlet and cable modem), can still be accessed.  I would like to
know how this is possible.
I am aware that some systems have "wake on LAN" modems, but I feel
that if a system is shut down and powered off, it will not be "visible
to the network (not broadcasting an IP address)" and also will not be
able to be accessed, because the hard drives/buses/OS are not running.
 The CMOS battery (in my opinion) cannot provide enough power to
facilitate hard drive activity.
I am not looking for extensive research, just a logical explaination
of how this may or may not be possible.  I am computer literate, so
dont be afraid to use the requisite jargon.  I have searched myself,
but see too many conflicting opinions.
Thank you, and have a great day!

No, it can't...if you have some kind of network storage device that is
powered separately, sure.  But the pc itsef? No way, no how.

Hi Frank, 
On the surface, I agree with you.  Someone recently told me they had a
relative who worked for a the FBI, and were able to do it.   I cannot
validate the claim, and I am hesitant to believe it, but I'd like to
find a logical answer on how (if at all) it can be done, or a response
the the person proving how it CANNOT be done.
Thank you!

Without Wake on Lan technology a turned off computer is going to draw
2.5 watts...a hard drive needs 5-10 watts to work and without the Wake
on Lan there is not going to be a way of getting more to the drive.

Pull out the 230v plug and it is dead to the World.

Take the battery out of a car and the ignition key is just tin.

If you are worried then pull out the DSL connection, preferably at the wall end.

That FBI stuff is probably hacking WiFi.

A computer can't be accessed when it is off. So in order to access a
computer, it must be on.  There are few ways to turn on a computer
other than pressing the power button (only some motherboards support
this feature):

1. By Dial up Modem - if a computer is connected to a phone line, it
will turn on when someone calls in when the computer is set to "turn
on by modem". the modem is always waiting for a ring, or a package
data that send from phone company to your phone. Once the modem
receive it, it will notify the motherbaord to turn on the computer. by
default this feature ususally is disabled.

2. By Keyboard, when any key from a keyboard is pressed, the cumputer
will turn on if it is set to "turn on by keybard", by default, it
usually is disabled.

3. Wake on Lan - this is so similar to the first one. But this one
uses an IP instead of a phone line, or phone #. In order to use this
feature, a mohterboard must have a wakeup-link that connects to the
network card.  the network card is always waiting for magic package to
arrive.  Once the card detects it, it will see if it is valid, if yes,
then the motherboard will turn on the computer.

First of all NEVER leave your computer unattended on the internet without a
firewall (you mention that possibility in your question).

Now, wake on LAN is based on Ethernet. The device sending the magic packet 
must be on the same LAN (as the name says). At worst, if you connect directly
to the Net this is limited to your ISPs machines. If you use an Ethernet 
connection to your modem then you have a local LAN with just your PC and 
your modem. If you have no other machines there, end of problem. It could 
be an issue in an office environment though, if another machine is hacked 
in the local LAN.

You missed the biggest problem though. The machine can program itself to turn 
itself on at a certain time. If someone has installed a malicious remote 
administration tool (RAT) then your machine can be remotely programmed to turn 
itself on at 2am (say) and dump important files in your hard disk at some
appropriate place on the net (for some details on RATs check
http://en.wikipedia.org/wiki/Remote_administration_tool )

This last strategy sounds like something the FBI (and also many hackers) have
the resources for, and I think is the most likely answer to your question.