Install zsh without root access?

If the machine has zsh on it you should be fine to just execute zsh on a prompt to swap, if that works you can either try chsh to swap your shell (should work without root) or when you're connecting with ssh put ssh -t hostname zsh to start it automatically.

You can try the chsh command. Are you sure zsh isn't installed on the distant computer? If not, you can do this (quoted from zsh FAQ Section 1.7)

The basic idea is to use exec to replace the current shell with zsh. Often you can do this in a login file such as .profile (if your shell is sh or ksh) or .login (if it's csh). Make sure you have some way of altering the file (e.g. via FTP) before you try this as exec is often rather unforgiving.

If you have zsh in a subdirectory bin of your home directory, put this in .profile:

[ -f $HOME/bin/zsh ] && exec $HOME/bin/zsh -l

or if your login shell is csh or tcsh, put this in .login:

if ( -f ~/bin/zsh ) exec ~/bin/zsh -l

(in each case the -l tells zsh it is a login shell).

If you want to check this works before committing yourself to it, you can make the login shell ask whether to exec zsh. The following work for Bourne-like shells:

[ -f $HOME/bin/zsh ] && {
        echo "Type Y to run zsh: \c"
        read line
        [ "$line" = Y ] && exec $HOME/bin/zsh -l
}

and for C-shell-like shells:

if ( -f ~/bin/zsh ) then
        echo -n "Type Y to run zsh: "
        if ( "$<" == Y ) exec ~/bin/zsh -l
endif

It's not a good idea to put this (even without the -l) into .cshrc, at least without some tests on what the csh is supposed to be doing, as that will cause every instance of csh to turn into a zsh and will cause csh scripts (yes, unfortunately some people write these) which do not call csh -f to fail. If you want to tell xterm to run zsh, change the SHELL environment variable to the full path of zsh at the same time as you exec zsh (in fact, this is sensible for consistency even if you aren't using xterm). If you have to exec zsh from your .cshrc, a minimum safety check is if ($?prompt) exec zsh.

If you like your login shell to appear in the process list as -zsh, you can link zsh to -zsh (e.g. by ln -s ~/bin/zsh ~/bin/-zsh) and change the exec to exec -zsh. (Make sure -zsh is in your path.) This has the same effect as the -l option.

Footnote: if you DO have root access, make sure zsh goes in /etc/shells on all appropriate machines, including NIS clients, or you may have problems with FTP to that machine.

Download zsh with:

wget -O zsh.tar.xz https://sourceforge.net/projects/zsh/files/latest/download
mkdir zsh && unxz zsh.tar.xz && tar -xvf zsh.tar -C zsh --strip-components 1
cd zsh

You can compile zsh yourself, for example:

./configure --prefix=$HOME
make
make install

and then start it explicitly, or programmatically from your current shell's startup file (put exec $HOME/bin/zsh -l in the right spot).