Is there like a server or something that I can run on a RPI (Raspberry Pi) that will ping a server every X minutes and if fails that it will send out an Email?

I don't really want an external service.

This might be overkill, but it simple and does what you ask.

 

https://oss.oetiker.ch/smokeping/

 

This is pretty overkill but you can use Zabbix or Nagios:

 

https://www.zabbix.com/

 

https://www.nagios.org/

 

I've used both (at work) and Zabbix is pretty easy to get up and running if you have some Linux experience.

 

I use an old Android phone running Tasker to do this.  Works great