I have an idea and do tell me if I’m being naive or not!
I want a website where customers can log online, and search by zip code for same day appointments close to them for hair salons.
For example- in the morning if a salon has a cancellation, they update their calendar to show there’s a last minute space.
The customer then goes on and searches by zipcode and that salon (and any others with spaces) pop up. They book the space and pay a deposit.
So both the customer and the business need their own profiles and calendars. The space will need to be blocked so no one else can book it and the booking needs to be visible in the customers profile.
It would need to handle a large amount of customers booking as well as hold a large amount of businesses.
Is this possible with a wordpress site? Can it handle that much traffic?
Any experienced people know roughly how much I would have to pay for a site like this? Any built a site like this?!

Handling the traffic isn’t your issue. You are asking the wrong question. The question should be “is wp the right tool for this type of infrastructure?” And the answer is no, it’s not.
You are attempting to cram a full fledged webapp into a CMS. It probably *can* be done to some extent, but not in a way that makes sense. For a single business wanting to offer online appointments? Sure. For a SaaS application for many businesses? Not so much. If you want a SaaS app build that. I see no value in forcing it on top of WP.
You could definitely build this with WordPress but the effort won’t be trivial (with any platform). But I would question the business case for this because (IMO) you may have some critical flaws in your assumptions.
There are many different scheduling systems that hair stylists may use (including paper calendars). They will probably not update their calendar in two places consistently if it’s a manual process. You’ll need to be able to integrate with and access/update their scheduling system via an API of some sort. Or convince them to move entirely to your platform. That will definitely be a challenge and in some cases impossible if their scheduling tool doesn’t have an API and the stylist doesn’t want to move entirely to your platform.
I would suggest interviewing 5-10 stylists and learning about how they do scheduling and new customer intake. Ask them about how their operation works and ask them about how they and their customers think about finding a new hair stylist.
Your also going to have to learn more about how devoted most women (and men) are to the person who does their hair and evaluate whether or not someone is willing to have just *anybody* cut their hair. I would submit that distance and availability aren’t their top concern often enough for the opportunity to put weight the development and support involved.
Again, IMO.