Are you booted into the Recovery Partition and trying to reinstall the OS? Have you ever downloaded the OS version you are trying to install using your Apple ID? If not, you won't be able to reinstall. If you are running El Capitan, try the link below when booted normally. You can make a bootable USB stick to install using this free program which will do all the work for you. You can then boot off the USB drive to reinstall.
Bootable USB Flash Drive – Diskmaker X oruse Apple’s method Create a bootable installer for OS
You can try to download it at the link below. If it won't let you, then your only options are to upgrade to Sierra/High Sierra or find an older computer where you can download it using your Apple ID.