How would you tell somebody that they're annoying you without being insulting?

I guess it's all in the delivery. Just come out and say it, but not in a threatening manner. Sort of casual like.

You don't tell people that. That is your problem. Be yourself and move on. 

I like to use off hand sarcastic comments that come with a certain level of implication.

Example: "Wow, man, you sure are talkative today."

I usually just avoid people that annoy me and lose all contact with them. I don't want to say anything because it never seems to come out right. I just avoid them like the plague!

Like others said, ignoring is best. If that's not possible because they won't stay outside your sphere then it needs to be addressed in hope it will give the better solution for both. Guess the more we try to go around it, the more awkward and annoying it becomes for both. Sober and directly, without too many words sweet talking it will often be the best way. Sober means no loaded words, there's no need for that. Hmm. Trying to tell me something? Ugh. This post was edited by Benedict Arnold at March 9, 2017 10:50 AM MST

Dear Dozy,
As far as the annoyance part, that I do consider my own problem, and my responsibility.
At the same time, I want to live my own life in my own way...

So I look for a tender and respectful way to get the situation moving.

* * *
Example: A dear neighbor whose (otherwise welcome) visit is just WAY too long. I might say, "you know, I've got a couple of irons in the fire here, really needing to move along, do you mind?"

" Love you to death man,   but you're annoying the patience right out of me."

just dont hang around them and then you wont have to say anything

I try to remove myself from the situation. If I can't I ask myself, "is this.person acting out of malice or are they just chatty?
If they are just chatty, I suck it up, and say nothing.
If not, and they're just being a Douche, then I tell them. Straight up.