I'm not that fast but I am at least locally competitive (24/F and run 18:40/38:56/85:48) and I didn't run at all in high school. I started freshman year of college.
I was already in shape from swimming, so my biggest mistake was taking things way too quickly. Pretty much as soon as I started I was running 40mpw, which I could actually handle alright, but as soon as I threw in a lot of quality (when I walked on the college team sophomore year) I got a stress fracture. I couldn't run at all at the start of that season and when I finally started getting back into it, I ran so poorly I got cut from the team.
I then spent the next year trying to walk back on the team. I did that by working really hard, getting hurt again, and repeating the cycle. I didn't get any faster.
Finally, my senior year of college I backed off and didn't race much and just focused on slowly building mileage. Once my mileage was higher, I cut it down a tiny bit and added workouts back in. Ran a few longer races, did alright.
During my fifth year I joined a club and started training hard again. This time, my body was actually ready for it. In a year, I dropped 5 minutes in my half-marathon and 1:20 in my 5K. I've been steadily improving since.
So my advice would be to treat the first four years of your running career like "high school." Take it slow, don't rush. But yeah it's totally possible. We have guys in our club who didn't even start until they were masters runners and they are damn good masters runners. You just have to be smart about it.