Don't let your age discourage you. I'm not sure how old you are, and if you don't want to tell us you don't have...I know it's safer for teens to keep their age hidden because of the way society is today. I'm going to assume you're probably mid-teens - 13 - 15? I'm saying that since you said people won't take you seriously at your age. Want some advice? Don't give up! :) Learn EVERYTHING you can about horse racing. Read articles online, go to the library and read books, keep posting on forums like this so us older, experienced riders/handlers can answer your questions about racing and racehorse training. The more you know, the more serious you will come off as to a barn owner or stable manager. What you'll have is good "book smarts", all you'll need after that is hands on experience. Ride, ride, ride as many types of horses as you can. Exercise riding and race riding, while quite different from any other discipline in the horse industry, still needs a confident, calm, experienced rider.
I know you'd much rather do something hands-on so if volunteering at a barn is out right now try this:
Do you ride your 13 year old horse English? Why don't you try taking your stirrups up a few holes and getting a sense of what it feels like to ride with your legs high up. The first time I sat in an exercise saddle I almost lost my balance! It takes some getting used to that first time to find your center, your balance point. You more or less "perch" on their back. So hike up those stirrups and try crouching over your horse in a jockey position. :) Do this at a walk in the ring, then try a trot. You'll see it takes some getting used to.
I started working at a racehorse barn as a hotwalker/groom when I was 16 and I know other people who have started working with racehorses young too. Just learn everything you can and don't give up! :)