There are a couple of issues with your script, but the most pressing one is the lack of then after the second test.
You also have an unnecessary test for $age -gt 18, which additionally introduces a logical error.  You already know that $age is less than 18 at that point, and you accidentally leave out the case of $age being precisely 18.  It would be better to remove the $age -gt 18 test altogether.
The script will not need the explicit call to exit at the end but should have a #!-line at the top pointing to the appropriate shell interpreter (probably bash).
You may find https://www.shellcheck.net/ a helpful site for finding the most basic errors in scripts.
#!/bin/bash
age=30
if [[ $age -lt 18 ]]; then
        echo 'not eligible'
elif [[ $age -lt 60 ]]; then
        echo 'eligible'
else
        echo 'stay home'
fi
     
    
https://shellcheck.net, a syntax checker, or installshellchecklocally. Make usingshellcheckpart of your development process.$age -gt 18... the firstifstatement takes care of that$age -gt 18 &&would make a difference ifageis 18, but probably your suggestion is the correct one instead of telling the 18 year olds to stay at home with the over-60ies …