I've successfully extracted the Fedora/OpenSUSE RPM into my home directory and ran chrome from there. You simply need to make sure that the symlinks for the libraries are all there. This assumes that the libraries area already installed, and $HOME/bin is in my $PATH.
I just ran:
mkdir ~/chrome; cd ~/chrome
rpm2cpio ~/Download/google-chrome-stable_current_x86_64.rpm | cpio -id
ln -s /usr/lib64/libnss3.so libnss3.so.1d
ln -s /usr/lib64/libnssutil3.so libnssutil3.so.1d
ln -s /usr/lib64/libsmime3.so libsmime3.so.1d
ln -s /lib64/libplc4.so libplc4.so.0d
ln -s /lib64/libnspr4.so libnspr4.so.0d
ln -s /lib64/libbz2.so.1.0.6 libbz2.so.1.0
ln -s ~/chrome/opt/google/chrome/google-chrome ~/bin/google-chrome
Now, if you don't have all those libraries installed already, or there are other dependencies for the chrome binary that are unmet, you might need to build and install them in your homedir. Google Chrome helpfully adds ~/chrome/opt/google/chrome/lib to the $LD_LIBRARY_PATH, so you could install those additional dependencies there.