Is there an ATC radio backup?
There is always a backup radio system as well as a several backup systems like for example a backup CWP (controller worker position) in case the primary fails. PANS-ATM (ICAO doc 4444) in chapter 2.5 SAFETY REVIEWS mentions that


  Operational and technical issues to ensure that: [...] communications, navigation, surveillance and other safety significant systems and equipment [...] include adequate backup facilities and/or procedures in the event of a system failure or degradation.


Emphasis is mine.

Also there is a standby generator (or even more than one) that has to instantly take over in case of power supply failure but I can't find any regulations at the moment.

As for the "what happens if the backup fails as well" question, we need to see the fuller picture that the document provides:


  d) communications, navigation, surveillance and other safety significant systems and equipment:
  
  1) are tested for normal operations on a routine basis;
  
  2) meet the required level of reliability and availability as defined by the appropriate authority;
  
  3) provide for the timely and appropriate detection and warning of system failures and degradations;
  
  4) include documentation on the consequences of system, subsystem and equipment failures and degradations;
  
  5) include measures to control the probability of failures and degradations; and
  
  6) include adequate backup facilities and/or procedures in the event of a system failure or degradation;


Emphasis is again mine

As you can see backup is the last line of defense. An ATS unit should ensure that equipment is regularly maintained, it's monitored, it's tested and any failures should be prevented if possible and their impact being documented.

If everything else fails, then you use the backup. And again the "include adequate backup" phrase can be translated as "if one thinks that a second backup is needed they should install one". But the previous 5 clauses are there to eliminate that case.