Short answer: Yes, you can do it!
Longer answer: Couch to 5k (c25k) really is designed to help people build the endurance to run either for 30 minutes OR for 5k (not both at once). If you browse around, you'll see success stories from people coming from all sorts of situations. Some people are once-fit people who want to be in slightly better shape, others are facing substantial weight-loss for their health. The one common thread is that if you apply yourself and take advantage of the support that's available here (and hopefully from your friends and family), the sky's the limit!
It's important to remember that the program is only a guideline. If you find that it's not working for you, feel free to tweak it a little bit. It's written to be competed in nine weeks, but many of us (including me) have repeated weeks. There's nothing at all wrong with that.
If you do decide to start the program, you do need to make one investment: properly fitted running shoes. That means heading out to a local running specialty store (NOT a shoe store, sporting goods store, or Foot Locker type of store) and having them take a look at how you stand, walk, and run so they can see how your feet will respond. That determines the type of shoe you should be wearing. Wearing the wrong shoes is a fast-track to injury and general unhappiness. It might cost you a little more (maybe not), but it's absolutely essential.
You might find that the program starts at a level you can't manage just yet. Again, no problem. Back it up and just walk for a few weeks, until you feel like you can give week one a try.
That's a start. I'm sure others will chime in with their own thoughts. If you're so inclined, see about linking up with some other people on the forum who are just starting the program around the same time as you. It's amazing how motivating it can be to share the experience with others who are in the same boat. I'm still in touch with several of the people who completed the program at the same time as I did (we post in the "Long Haul" thread; you're certainly welcome to join us there), and we still support each other.
(Edited to add: Just be careful about any medical conditions you might have. You didn't mention anything that seems like it'd be a problem right away, but it's a good idea to be careful! Get clearance from your doctor if you have any concerns.)
Welcome to the forums and to the beginning of your life as a runner!
Don
P.S. Not to burst your bubble, but not every run is zen-like. Some of 'em downright suck.
The good ones more than make up for it, though. 
"Don't fear moving slowly forward ... fear standing still" - Kathleen Harris
"Last is just the slowest winner." - C. Hunter Boyd