Can I run multiple instances of Dropbox at once?

Yes, this should be possible. Here is a snippet from the gentoo dropbox init script:

 start() {
     ebegin "Starting dropbox..."
     for dbuser in ${DROPBOX_USERS}; do
         start-stop-daemon -b -o -N ${NICENESS} -c $dbuser -S -v -e HOME="/home/$dbuser" -x /home/$dbuser/.dropbox-dist/dropboxd
     done
     eend $?
 }

In your case the commands you need to run would be

HOME="/home/user/.dropbox/work" /home/user/.dropbox-dist/dropboxd
HOME="/home/user/.dropbox/personal" /home/user/.dropbox-dist/dropboxd

be sure to link the ~/.dropbox/work/Dropbox folder created on the first run of each process to ~/Dropbox/Work or ~/Dropbox/Personal or change it using the script described in the dropboxwiki.