When I finally have enough archers to take out that barb camp
A single warrior can take out a barb camp
A slinger and a warrior can take it out fast because the barb will chase the slinger
It is purely about stopping a scout spotting a city and getting back to camp.
The rest is not so tricky.
I just started a game on emperor and on turn 11 there is a barb camp 4 tiles from my capital with two horsemen that killed my scout. and that is apparently par for the course for this patch.
after many many test start I can say with confidence that a barb camp 4 tiles from your city is a Very rare beast unless You are packing extra civs and CS in. 1 camp within 4 tiles over 8 civs happens perhaps once per 4 games but naturally on higher difficulties there is less places to spawn. It happens, but it does not happen every game by any means and often not to you but one of the civs near you which helps you a lot. If you have 8 civs the first 20 turns of the game (Roughly) spawn camps at most, then they stop apart from quite random ones.
I recommend you keep your scout/sling/warrior fending off barb scouts for the first 20 turns and keeping that visibility around your city up. You lower the odds hugely.
Even when a camp spawns within 4, the scout starts on the NE tile of the camp and moves randomly so can completely miss your city.
Putting your city on a hill Hugely increases the chances of a barb raid against you. The scouts can see your city from 3 tiles then rather than 2 and can see It over woods and hills (but not wooded hills) They need to see your city, not just the borders like other civs do.
You can take the approach that barbs no longer will destroy your capital so you can ignore them but once your city is spotted you are the target of a barb spawning camp for about 50 turns.
What's the rule for how fast camps spawn units? Sometimes they pump them out faster than I can kill them for a while.
on immortal or deity it is 1/turn but more units at deity. Below it is every 2 turns and less units with every level. They spawn in 2 sets. A set to raid you which times out after roughly 20 turns and then they city attack you with another set if any of their raid troops got back to camp. A city attack differs in it has siege units and lasts roughly 30 turns.
they spawn an archer when half the civs have archers... and so on. So they get support and flanking when half the civs have military tradition. They are a horse camp if there are horses within 3 tiles but these are weak horses until half the civs have horsemanship.
You must keep up with tech or fog bust, doing neither means you will hate barbs big time.
Early army means less barb hassle and more chance of a golden classical era unless you have a few CS nearby why heavily target barb camps.
Civs that are not coastal are much more vulnerable to barbs although an early barb quad is unpleasant.