How can I check that a remote computer is online for ssh / script access?

Like this:

nc -z hostname 22 > /dev/null
echo $?

If it's 0 then it's available. If it's 1 then it's not.

Use netcat:

nc -z localhost 22

From the manpage

 -z      Specifies that nc should just scan for listening daemons,
         without sending any data to them.

What about

MACHINE=192.168.0.8
exec 3>/dev/tcp/${MACHINE}/22
if [ $? -eq 0 ]
then
    echo "SSH up"
else
    echo "SSH down"
fi

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