No, it's not too crazy. It's only a little crazy, so that's all right.
You need to get an OK from your doctor first.
Do you belong to a gym? If you do, start working on your stamina as well as your strength. Those 3.1 miles are going to seem as long as a marathon course at first!
If you don't have a gym, you can still plan and train. Check out the C25K web site. They will give you reasonable assignments for each day. If you need to modify them, that's all right; it will just take a little longer.
You also want to look at what you're eating. You need healthy foods to give you the energy you'll need. Don't be afraid of good carbs.
What kind of shoes do you have? You need good athletic shoes AND good socks. They don't have to be the most expensive, but they have to be good quality. Is there a store in your area where they can look at your feet and your gait and recommend what would be best for you? (I'm basically a thrift-store person for clothing and shoes except for my athletic shoes. Then I'm willing to spend up to $100.00 - which isn't expensive, as you'll find out.)
Don't try to be trained in a week! Your children need you alive. Slow and steady is the way. Get someone to help you measure a one-mile distance inside your gym (say, so many times around the perimeter of the basketball court) or around your neighborhood. Then try walking a distance just far enough to be a little work for you. Try to improve on that bit by bit until you've worked up to a mile. When that seems comfortable, add on to it. If you can keep track of what you do every day, you will know when to reward yourself. Always take water with you when you practice.
Look online for a local runners' group that posts races coming up in your area. You can also ask where you buy your shoes. And... it's perfectly fine to WALK 5ks. It's a great way to start. You can add jogging/running as you can. A young mama friend of mine does her outdoor training with two children in a jogging stroller, which is great because it adds a bit of resistance (just a bit?).
For what it's worth, I'm much older than you are, and about as naturally active and athletic as a marshmallow, but I started doing 5ks and an occasional 10k three or four years ago. I'm out of it now because of an injury, but I hope to get back to it next spring. This pastime is definitely habit-forming! And it's fun. I feel SO good when I've finished a race (even though I'm always in everybody else's dust). And it's great to see moms and dads doing it with their children.