My Mac has no operating system and it won’t start up in recovery mode

Try booting into Internet Recovery (command - option/alt - R on a restart). It is a slow process. This will install the newest OS that the computer is compatible with. Reformat the drive using Disk Utility/Erase Mac OS Extended (Journaled), then click the Option button and select GUID. Quit Disk Utility. Then install the OS. When the computer restarts, you can use Setup Assistant to restore your data from Time Machine.

OS X Recovery

OS X Recovery (2)

Do you have access to another Mac computer where you can sign into the App Store using your Apple ID and download the OS. Then quit the installer.  You can make a bootable USB stick to install using these free programs which will do all the work for you. Boot the computer with the option/alt key held down and select the USB drive. Run Disk Utility and select First Aid. Then re-install the OS.

Bootable USB Flash Drive – Diskmaker X      Disk Creator   or use 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/Mojave or find an older computer where you can download it using your Apple ID.

El Capitan

How to upgrade to macOS Mojave - Apple Support

How to get macOS High Sierra