What it is
The classic “alt-ac” path: helping colleges and universities operate by performing any number of functions other than teaching or research. Academic administration can be divided into several basic areas. The three most relevant to STEM PhDs are Academic Affairs, Student Affairs, and Information Technology. Academic Affairs centers on curricular and programmatic issues. Student Affairs is concerned with the whole student, providing a wide range of services and resources. Information Technology supports all university activities through the selection, implementation, and management of technology.
Any staff position that does not require faculty experience is potentially a good starting point for a university-based career. Staff positions become available year-round as people leave or as new positions are created. Your main selling point will be skills, but familiarity with a particular campus is helpful in that campus (to all applicants, not just PhDs). More staff positions require PhDs these days, particularly in Academic Affairs, so that may work in your favor. The PhD is certainly not viewed negatively in higher ed.
Once in, you can switch units relatively easily to follow your interests and eventually move up to Director of your favored area. You can move to a different institution much more easily than you can as faculty, or can even move to another type of institution, such as a hospital or government agency. Salaries are about the same as faculty salaries and can be higher once you move beyond the entry level. Benefits are usually excellent.
All STEM disciplines can potentially enter administration. Some staff positions require subject matter expertise, giving PhDs in that subject an advantage, but even then, skills and experience trump subject matter knowledge.
Personality and outlook
Because there are so many different kinds of positions, it is safe to say that almost anyone can find a pleasing role in university administration. Consider the types of things you enjoy doing, and look for staff roles that include those activities plus others you’d like to learn. Social and “soft” skills are important in all positions, more in some than in others.
As a grad student or postdoc, pay attention to the wider university beyond your own lab or department. A modern university is a complex organization, with many different ecosystems and cultures throughout. Study the university website to learn the units; drop in on various offices; conduct informational interviews to get a sense of what people do. Seek out part-time positions or internships in units that interest you or where you have an existing relationship. Gravitate towards service assignments that will expose you to other facets of the university. Take a class on higher education administration or something related. Develop non-academic skills!