I'm probably echoing other people's sentiments to a high degree, but that never stopped me before.
I have a degree in computer science, but it really means nothing in sysadmin work other than adding some more theory of "why things in a starship work." (If you know the reference, that may be a good sign.)
In my experiences college does little more than give a benchmark for employers to think you are qualified for the field they themselves can't judge (if they're not technical) while telling other techs in a business that you paid some dues. But the reality is that theory doesn't always hold up when you get into the field.
There are things done in various fields that are more a product of politics than technology, and theories break down fairly quickly at that point. You'll be asked to do things that are completely contradictory to good practices. You'll have users that are exempted from security procedures. You'll have to come up with "workable" solutions that you know are stopgaps or poor solutions, but as long as certain others are happy with it, it's going to become permanent.
Human communication is also often lacking as a skill. I sometimes think that a sysadmin needs a team of two; a person that speaks to users and bosses, and a person who knows the technology and keeps it running, because I've rarely found people that excel at both. It would be nice if all the sysadmin duties were carried out from a hole in the basement populated by Leatherman-armed flashlight-carrying ninjas and all user interaction with the Sysamin dungeon were mediated by the communications-gifted front desk. Users would probably appreciate it too. They tend to not like their IT people. We speak funny words and phrases that they not only don't understand, but don't want to understand. And we make them feel stupid. Why would they like us in that case? And the only time they want to talk to IT is when something is broken. We see them and hear from them when THIS MUST BE WORKING RIGHT NOW BECAUSE IT'S THE MOST IMPORTANT THING IN THE WORLD THAT I GET THIS DONE AND THE INTERNETS ARE BROKEN.
So what prerequisites do you need?
Ability to speak human.
High tolerance for learning.
Ability to put aside technology biases long enough to use the right tool for the right job
Ability to recognize when your job environment just isn't for you. There is no set "sysadmin" job. Fields hold similarities, but one sysadmin job may be hell while another gives you more respect but both have you doing the same thing; this is especially true for a field where your bosses may have no clue what the hell you really do.
Ability to recognize you are not your users. Your users don't give a @#% about VOIP, QoS, routers, antiviruses, disk fragmentation, network shares...they just want to get the job done and focus on tasks. I remind myself of fields I suck at and pay others to do for me because I don't care how it's done either when I realize I'm resenting my users. I just want my car's maintenance schedule done, I just want the plumbing to work, I just want my taxes paid so I don't go to jail...ideally, your users won't know that you're keeping them running unless you get hit by a bus and things fall apart.
You need to document things clearly. If you were hit by a bus, your fellow techs or your replacement shouldn't need a huge amount of time to figure out what you rigged together for backups, how you were imaging things, what you're doing for system monitoring, etc.
You need to learn the arcane art of google-fu. Big trade secret. Since everything changes so rapidly, you need to keep your skills up to date using the googles.
You need to learn to balance the stuff that's important to other sysadmins with stuff that's important to your employers. Okay, this one's debatable. What I mean is that the geekery in sysadmining tends to downplay certs and such; they're paper that means something to employers who aren't geeks and the skills they certify you for tend to be something that changes rapidly so they're outdated by the time you get them framed, unless it's something like putting ends on fiber patches or something like that. Sysadmins are in a hierarchy of geekery that is based on a meritocracy. The end result is that a sysadmin in charge of other techs needs to gain respect from his charges by doing, and learning, and immersing in the ever-changing field, while if he or she wants to advance in the politics of the business he or she must continue to do things that others can use as a benchmark when they don't understand much at all about what you do. It's stressful.
Know your limitations. A sysadmin may be very good at infrastructure, or VM configuration/management, or backups, or databases or any of a mixture of things. But if you don't know how to do something then ask or find help. There's nothing shameful about an admin who doesn't know everything about something that has electricity and a power plug (which believe me, your users will think you're in charge of it if it plugs anything into the wall.)
Play. Play with tech at home and on the job. The best sysadmins I've run into are people who didn't get into it because one day they thought computers would be neat or profitable. The best sysadmins were people who spent time playing with Linux because it was a neat puzzle that gave them insight on how the system worked. They read articles online about how RAID worked, and maybe played with software RAID. They learned the arcane lore of how boot sectors worked, and spent hours recovering data from corrupted drives not because the backup wasn't made (because the best sysadmins HAVE GOOD BACKUPS) but because they learned in the process a little better how things worked. They understand why things on a starship worked. It helps them troubleshoot, it helps them understand the best solutions to a given problem, and it helps them understand their options when confronted with an issue. The best techs are the ones that don't just follow checklists. They make the checklists and know why the items are there, and if they don't know why, they make time to question it.
You should learn to share your knowledge with others. Post it online. Contribute to groups, and on places like Serverfault. You're going to require a lot of googling on the job and you're benefiting from other people's experiences, and the information they share. Give back whenever you can. No geek or techie appreciates people who hold on to knowledge in an organization to keep their job (i.e., you can't fire me because I'm the only person that can run the backups! HA HA HA!)
No doubt others have had other experiences, but these are the things I've run into. Some of it is probably biased and/or not in line with certain job profiles, but hey, I'm open to corrections and/or edits, and if this gets downvoted into negative numbers I'll probably delete it to keep from polluting the answer stream with unhelpful answers, but this is one sysadmin's experience, for what it's worth.