There is no technical number. It is dependent more on how many extra fields there are for a user since WordPress stores these as individual user meta. More meta fields per user equates to more server load.
The most I know of is someone running a site with over 150,000 users, but there are only four extra meta fields per user.
Chad is right. The number of people that can register on a site is directly based on the size of the database itself. The size of the database is determined by the type of account you have with a hosting provider.
The only viable solution to ensure a max number of registrations is allowed is to monitor the amount of people who register and when the number is met, remove the registration page.
