Learn Italian: Ho preso una precauzione drastica sul balcone. - I took a drastic precaution on the balcony.

Yes, Italian perfectly mirrors English here with the collocation prendere una precauzione. You will also see prendere used for similar abstract actions, such as prendere una decisione (to make a decision) or prendere un provvedimento (to take a measure).

Why do we use the preposition su (sul) for the balcony instead of in?

Italian treats a balcony (il balcone) or a terrace (la terrazza) as an open exterior surface rather than an enclosed space. Because you are standing on a surface rather than inside a room, it requires su, which combines with il to form sul.

Could I put drastica before the noun, as in una drastica precauzione?

You could, but it changes the tone. Placing drastica after the noun is the standard, objective way to describe the type of precaution. Moving it before the noun (una drastica precauzione) would make it sound more literary or add strong subjective emphasis to how extreme your action was.