Ask HN: As a contract to hire for a startup, how do I obtain equity?

You don't. The whole point of working on a contract is that you trade a (substantially) higher bill rate for all the standard benefits of working as an employee.

So no healthcare, no company provided machine on your desk, no paid vacation or sick leave, no retirement contribution, no invite to the xmas party, and no stock options or equity.

Negotiate your contract with that in mind, and charge accordingly. When and if you decide to do the "to hire" part of the arrangement, negotiate that bit knowing that equity and free cake at Stan's birthday party are now back on the table. (Favor the latter, as it's likely worth more).

Stock and stock options are also usually a means to keep an employee tied to the company (less likely to leave). I've done bonus payments at certain events, e.g. financing rounds (all written in contracts of course) and seen companies giving a fixed amount of stock to advisers. But a contractor owning any significant equity or stock options doesn't make sense for the employer.

This may sound a bit short but, ask a simple question, get a simple answer. You'll want to negotiate that as a part of the hiring process as you move from contract to hire. Just as you would do with a purely hiring situation.

Do you have some further concern about making this happen?

get converted first, bargain then.

Ask for it.