How old do you have to be to do olympic archery?

there are no age limits in Archery or any olympic sport.

the limiting factor is being able to hold and use the equipment sucessfully and score the points at the required distance.

I am a GNAS archer and former instructor but can never get better than Bowman 2nd class as my size and strength means I cannot fire the required number of shafts (arrows) over the distance needed to get to 1st class.

try it and if you get to qualifying standard ... go for it