O script que você quer é algo assim:
#!/bin/bash
# by desgua to test server down
#
# This script is supposed to be named ".ping.time.sh" and
# to be put at ~/
# to test what happens when server is down:
# change www.mysite.com for www.serverdown.com
#
# As asked, I supposed you have a script to send mail and named
# ~/.script.to.send.mail.sh
#
sleep 300 # as asked, to run at every 5 minutes
pingtime=$(ping -w 5 www.mysite.com | grep ttl)
echo $pingtime
if [ $pingtime = ] ; then
echo 'down' ; ~/.script.to.send.mail.sh ; else
echo 'up' ; ~/.ping.time.sh
fi
ps. Em seu script para enviar o e-mail, coloque algo para executar novamente o script que fiz depois que o e-mail foi enviado