Is there a scientific formula for how close one can get to a rainbow?  

Rainbow is not an object like the sun and the moon. It is an apparition. It is not easy to give a formula for the distance. Each colour comes to you at a particular angle. The source of light, the sun has to be  sending the light for the water drops for total internal reflection to take place.

The rainbow need not be always far away, can be at your arms distance if you spray some water and the is sending light at the correct angle. The distance to the will depend on many factors such as 1. the sun's position, 2. size of the water droplets. 3. The shape of the drops depending on speed of fall the air resistance, 4 Temperature and so on.

 There are far too many factors to be considered. We hope to include a lesson on rainbow soon describing how it is formed. 

 Thank you very much for sending the question.