I get error 403 on my old iMac with El Capitan after reinstalling the OS

Erasing the computer without another startup disk in hand was a mistake.

You may not want to hear that, but it was.

These days, with old versions of Mac OS X, it's always best to have a bootable backup to work from when you are thinking about erasing your main startup disk. With a bootable system, you could do things like

• Downloading the disk image containing the El Capitan 10.11 installer, mounting the disk image, and running the installer (against the target drive).
• Cloning the bootable backup back to the main drive, in the event that you couldn't get the old OS you needed from Apple's servers. I have seen other reports of Internet Recovery not working for old versions of the OS … enough reports that I would not want to be completely dependent on it.

The El Capitan 10.11 link is from How to download and install macOS - Apple Support . I copied and pasted it here because the latter link is a https one, and https security has changed over the years in ways that prevent a lot of old versions of Mac OS X from being able to access https sites.

You may need to find someone else who has an old Mac (maybe a repair shop in your area) and get them to fix up a bootable external drive for you, to get your system going again. Whether you could upgrade to a version of the operating system newer than El Capitan would depend on your specific Mac model and hardware release year. I'd normally ask someone to get that information from About This Mac in the Apple () menu, but you probably will not be able to do that until such time as you get the Mac running again. All we have to go on now is that the Mac was running El Capitain (which we know was the end of the line for some models, and not for others).

As expecting rkaufmann87 to solve your problem instantly (without any help from you), or to buy you a new Mac, both expectations are unrealistic.