How do I say no to pushy people trying to reach my boss?

You can gatekeep now. The ordinary requests you handle fine. It is the aggressive ones who will not take no, and the important people your boss might actually want but did not ask for, that still put you in a bind. Those judgment calls are the hard part, and they carry real risk. For the aggressive ones, remember that their pressure is not your problem to absorb. A firm, calm no does not require their permission. You are not being rude. You are doing exactly the job you were given. Their frustration is the friction of the wall working, not a sign you did wrong. For the important-but-uninvited, when you truly cannot tell, ask your executive rather than guess. A quick note, this came in, do you want it, protects you both. Over time you will learn their preferences and guess less. The bind eases when you stop trying to be certain alone and let the one clear rule be: when in doubt, check.