(so with the exclaimer that I am also a newbie to peacocks, but I have raised chickens)
Nothing more than 1/2" hardcloth from everything I have read. It depends on your predators. I have everything from Coyotes, Foxes, Racoons, Ringtail Cats, Opossums on our ranch and all active. If you are expecting issues they say 1/4". And do yourself a favor and don't skimp on the price. For me I will do 1/4" pvc, galvanized on the shed and on a run (still debating in my head) 2x3 to 50" with a 18" predator skirt (because there is no digging into the bedrock we have), electrified (I think) on the outside of the run with a netting on top (especially recommended for birds that flush) that I will go up to about 10 feet with. My shed is having to be built and it is off a
greenhouse plan, so I can use it for that later.
For penned peacocks I have seen anything recommended from 100sqft per bird to 150sqft per bird.
But like you, I intend to free range here, and not specifically breed them. So you can end up with a situation where you are thinking they just need temporary lodging (sorta). The question is when do you expect to release them (6 months to nine months for me), and in that mean time how much space is needed for the age they are at, at a specific time of year, etc. But as a wise person told me a few weeks ago ...
I asked a question on brooder bottom options, and KsKingBee had some good insight and pictures of what they do.
https://www.backyardchickens.com/threads/brooder-bottom-options.1665939/
Sorry for the length of this, and that I am not more experienced with peafowl. Currently, I am reading everything I can get my hands on, hours a day. The wealth of info comes from the people who have done this the longest. I feel like the best thing I can do is read, and try not to ask too beginner a question. I know from my other hobbies, people appreciate it when you have done your homework. I was luckily to have been given good info here to start with, and that sticky thread has been a lifesaver a couple of times. I have some eggs arriving from a neighbor tomorrow to try to hatch <<crossing my fingers>>. I am so glad you didn't have any issues. I don't expect to be as lucky.