Boot into recovery mode using Command + Option + R which should not require authenticating using an AppleID.
You will also likely need to change the date on the computer to some time in 2017 or perhaps even earlier since the certificate for the installer has probably expired. While booted to the installer launch the Terminal app from the Utilities menu and use the following command to set the date & time to Jan. 2, 2017 @ 11:33am (adjust the year if necessary):
date -u 0102113317
If there are no errors, then quit the Terminal app and select "Reinstall macOS" unless you are performing a clean install which will require you to first use Disk Utility to erase the drive before selecting the "Reinstall macOS" option.
FYI, if you can still boot the Mac, then I highly recommend you create a bootable macOS USB installer using the instructions in this Apple article:
How to create a bootable installer for macOS - Apple Support
If your Mac can run macOS 10.10, then it should be able to run macOS 10.11 as well. You should check to see whether your Mac is able to run an even newer version of macOS by using this article to see which versions of macOS are compatible with various Apple hardware:
https://eshop.macsales.com/guides/Mac_OS_X_Compatibility
If for some reason you need macOS 10.10 specifically, then here is a link to the installer. You can still create a bootable macOS 10.10 USB installer by following the directions for the 10.11 El Capitan and adjusting the command accordingly to reflect the difference in names.
How to get old versions of macOS - Apple Support