For the moment, I've decided to continue using Ruby from the Windows installer provided by rubyinstaller.org.

When using this method, you have to install bundler separately using the command:
gem install bundler

I did not get the bundler stack error you noted during gem installs, nor did get the compass error you identified. Both those are probably path/cygwin implementation issues.

I did, however, get a different error during bundle install:

Gem::InstallError: The 'oily_png' native gem requires installed build tools.

Please update your PATH to include build tools or download the DevKit
from 'http://rubyinstaller.org/downloads' and follow the instructions
at 'http://github.com/oneclick/rubyinstaller/wiki/Development-Kit'
An error occurred while installing oily_png (1.1.0), and Bundler cannot
continue.
Make sure that `gem install oily_png -v '1.1.0'` succeeds before bundling.

I followed the directions in that error message and installed the DevKit (the installation commands are provided in the GitHub wiki note), and then re-ran bundle install and everything seems to have worked.

I'll be curious to see if not having rvm actually makes much real-world difference in the case of small-scale, single-coder, local development.

Also, it appears there is an alternative to rvm for Windows developers called "pik" and I wonder if that might be a better option than rvm via cygwin?:
https://github.com/vertiginous/pik

Added 2 May 2014:
User Stickton has written up instructions for using rubyinstaller and pik here:
Omega 4 Sass Quick Setup in Windows

OK, I got the message:

ruby-1.9.3-p484 is not installed.
To install do: 'rvm install ruby-1.9.3-p484'

...But after trying to run that command line, I get:

$ rvm install ruby-1.9.3-p484
Searching for binary rubies, this might take some time.
No binary rubies available for: cygwin/unknown/i386/ruby-1.9.3-p484.
Continuing with compilation. Please read 'rvm help mount' to get more information on binary rubies.
Checking requirements for cygwin.
Requirements support for cygwin is not implemented yet,
report a bug here => https://github.com/wayneeseguin/rvm/issues
Requirements installation failed with status: 1.

I've tried several ways but didn't work. And I finally succeeded by installing railsinstaller.
Maybe this is the easiest way for windows.
No rvm no cygwin or rubyinstaller,just railsinstaller.

As for "... stack level too deep (SystemStackError)...", the command "peflags -X409600 which ruby" gave me the following output in my cygwin Terminal:

$ peflags -X409600 which ruby
which: skipped because nonexistent
ruby: skipped because nonexistent

The solution was to change it into:
peflags -X409600 `which ruby`

It looks like rvm has been updated to work with cygwin. There is a long conversation at https://github.com/wayneeseguin/rvm/pull/2360#issuecomment-32062384.

The bottom line for me was to get the current version of rvm. Run from cygwin root,

rvm get head

and then

rvm requirements.

This would give errors like:

Error running 'requirements_cygwin_libs_install sqlite3 mingw64-x86_64-gcc'

Install any missing items (sqlite3 and mingw64-x86_64-gcc) using the cygwin setup.exe and you should get the message:

Checking requirements for cygwin.
Requirements installation successful.

From the subtheme directory, run

rvm install ruby-1.9.3-p484

I needed to run

peflags -X409600 `which ruby`

as mentioned above to increase the stack size before running

bundle install

I get through most of the gem install with the Bundle Install comand but after about 12 gems are installed I get the "stack Level too deep" error. I ran

peflags -X409600 `which ruby`

but that only took me from no gems to about 12 gems installed. Any one have an idea how to solve this?

/ruby-1.9.3-p484@global/bin/ruby_executable_hooks:15: stack level too deep (SystemStackError)

I tried to install all of this on my laptop Lenovo Z580 and the same errors (space is occupied in screenshot) are still happening.
I use XAMPP and Avira Antivirus on both computers could these be the culprit?

Edit: Ran everything in safe mode and still got the space is occupied errors :(

I am also getting lot of fork errors on my windows 7 machine. I ran rebaseall but it didnt help. Please help.

After entering
bundle install
I get the "stack level too deep" error:

Fetching gem metadata from https://rubygems.org/..........
Fetching additional metadata from https://rubygems.org/..
Resolving dependencies...Unfortunately, a fatal error has occurred. Please see the Bundler
troubleshooting documentation at http://bit.ly/bundler-issues. Thanks!
/cygdrive/c/Users/BaSe/.rvm/gems/ruby-1.9.3-p545@global/bin/ruby_executable_hooks:15: stack level too deep (SystemStackError)

but running
peflags -X409600 'which ruby'
results in
which ruby: skipped because nonexistent

I followed the instruction as mentioned above, however I am getting following error.

An error occurred while installing ffi (1.9.3), and Bundler cannot continue.
Make sure that `gem install ffi -v '1.9.3'` succeeds before bundling.

The details of the log is here http://pastebin.com/0bJ7JREy

I went to my subtheme's root directory in Cygwin and RVM told me to install ruby 1.9.3-p547 so I did and everything went as normal however when I try and run omega-guard in Drush I get prompted that "You have to install Ruby version 1.9 or newer." I'm guessing I missed something really simple but not sure what I could have missed from the tutorial. Has this happened to anyone else?

Done, dusted and working nicely in 5 mins flat. http://railsinstaller.org/en

If you get an error like this (while installing the eventmachine gem):

make
compiling binder.cpp
make: g++: Command not found
Makefile:198: recipe for target 'binder.o' failed
make: *** [binder.o] Error 127

make failed, exit code 2

Gem files will remain installed in /home/Zach/.rvm/gems/ruby-1.9.3-p547/gems/eventmachine-1.0.3 for inspection.
Results logged to /home/Zach/.rvm/gems/ruby-1.9.3-p547/extensions/x86_64-cygwin/1.9.1/eventmachine-1.0.3/gem_make.out
An error occurred while installing eventmachine (1.0.3), and Bundler cannot
continue.
Make sure that `gem install eventmachine -v '1.0.3'` succeeds before bundling.

then rerun cygwin setup and enable the gcc-g++ package. (I replaced gcc with gcc-g++ in the documentation.)

I have been trying to successfully install RUBY, Bundler and then run the "drush Omega Guard" for days and I finally figured it out! Better late then never.

WINDOWS 7 32 bit:

First, you do NOT need cygwin or RVM to get it to work. The issue is with the newer versions of RUBY. Long story short, you must use RUBY 1.9.3. Follow the direction below:

1.) Download and install RUBY 1.9.3 from the http://rubyinstaller.org/downloads. (If you have a newer version of RUBY installed, be sure to uninstall it first)

2.)While you are at http://rubyinstaller.org/dowloads, grab the DevKit 7zip file. Be sure to get the correct version for RUBY 1.9.3. After you have placed you DevKit somewhere permanent, (FOLLOW THE DIRECTIONS for the DevKit here, https://github.com/oneclick/rubyinstaller/wiki/Development-Kit#building-...). Just start at "Installation Instructions".

3.)In the Ruby cmd, run gem install bundler. (FOR SOME REASON, I HAD TO INSTALL EACH GEM individually, it was about 47 of them. Didn't take that long, 20mins)

4.) Run Ruby cmd command, "drush omega-guard"