You can try booting into Recovery Mode using Command + Option + R. If you have a 2008 or 2009 model, then that may bypass the need to authenticate with your AppleID. If you have a 2010-2012 model, then it should give you access to the online macOS installer. If you can access the online macOS installer, then you may need to used Fix#3 in the following article since Apple has not been keeping their older online installers fully operational (at least for High Sierra....the "fix" should work for any installer which gives this error):
https://mrmacintosh.com/how-to-fix-the-recovery-server-could-not-be-contacted-error-high-sierra-recovery-is-still-online-but-broken/
What is the exact model of your laptop? You can try to get that information by using the Apple check coverage page here:
View Coverage - AppleCare & Warranty
If you have access to another working compatible Mac, then you can try creating a bootable macOS USB installer. Here are the general models needed for each installer:
- macOS 10.15 Catalina -- 2012 to mid-2020
- macOS 10.13 High Sierra -- Late-2009 to mid-2018
- macOS 10.11 El Capitan -- 2007 to 2015
You can verify whether the other Mac is compatible by using the information in the following article:
https://eshop.macsales.com/guides/Mac_OS_X_Compatibility
Some Macs up to & including 2010 models did ship with OS recovery DVDs or could use an OS X 10.6 Snow Leopard DVD installer. We would need to know the exact model of your laptop in order to tell you which exact DVD you would need. It is very complicated if you need to start with a DVD.
Also, why did you erase the laptop? If you are trying to solve a problem, then that problem could also cause you problems with reinstalling macOS. It would not surprise me if your laptop has a worn out or even failing Hard Drive.
