Its doable but its not going to be easy. If your going to create a whole booking system then you would want some way to store that via NocoDB or a GOOGLE Sheet. Then create an account per user or per group that can be managed from a typebot interface so they can adjust there availability. Everything in typebot checking the dates and times of people who want a booking. (its alot).
Take it a step further you could use cal.com api and use it as a webhook ran by typebot.
https://cal.com/docs/api-reference/v2/introductionIl not sure if the self hosted cal.com has api functions but i assume it does. From there you can just use typebot to check whos able to be booked or just let cal send a booking to whoever has a slot. (again its a bit of work)
No easy button unfortunately.