Generally speaking, students break into it via ISV (independent software vendors, the big ones in this space are IBM and Broadcom but there are many others) apprenticeship and training programs, before getting deployed on site (or staying at the software vendor if you are very good or they have need of somebody junior for a position).

The extreme experience requirements are generally exaggerations. They only happen to exist because there are people who really do have those many years in this field. But I've gotten jobs (I'm a systems programmer, many hats style, z/OS and Linux and C and JCL and Shell and REXX etc) and I've gotten jobs before from these sorts of excessive experience postings before despite me being fairly new to the field (just reaching senior level, about 5 years experience in my subdiscipline, which is systems programming, as opposed to applications programming or security, two other primary specializations for people with CS degrees, there is also operations and IT work available for mainframes but probably the pay rate would be of no interest to somebody on HackerNews).