Is it legal not to pay employees a salary (even not a minimum) and only incentivise them with an equity reward?
Do I, as the owner of the company, have to pay a salary for myself?
You've mentioned equity. If people joining the company are immediately given equity, they become owners (and possibly directors) rather than employees. This would mean that you could no longer consider yourself the only owner of the company. It's not a requirement that owners take a salary - though it's a good idea.
If you do not immediately give equity, there would need to be a contract in place showing how much equity your partners would receive, and when. This should consider the possibility of someone leaving the company before the first assignment of equity, and make arrangements for how they would be remunerated for any service to date. Again, they would be considered partners and part-owners and not employees, and the contract would be a partnership agreement and not a contract of employment.
Short answer : startups can't get out of paying the minimum wage to their employees, but don't necessarily have to pay their owners.