I feel like the authors of this article are being very misleading by using broader terms where they could be more specific, like using system where they could instead say software. The IRS isn't running 60 year old hardware, and the failure was in hardware that was just 18 months old. But because the IRS is running software that is 60 years old, the authors can claim the "system" dates back to the 1960s.

It's also clear the authors don't entirely understand what they're talking about. In the second article linked to, which is written by an author of this article, he states "[t]he Individual Master File, a massive application written in the antiquated and low-level Assembly programming language" as if to imply there is but one assembly language.

A GAO report I found chasing the last link in the article indicates that the two oldest systems in the Government are the IMF and Business Master File (BMF) which it says both run on IBM Hardware. My guess would be that the IRS is running modern z/Architecture mainframes in System 360 Compatibility mode. The System 360 dates back to the early 60s and IBM has maintained backward compatibility with it all the way up through z/Architecture systems. This would also explain why they're so far over budget and behind schedule.