Is it possible to do a cheap or free online or distance Ph.D. program? Is it possible do to an online or distance Ph.D. program at all?

If so, are there ones that are cheap or free?
To a first approximation, it's not possible.  There are few online or distance Ph.D. programs, and they usually have bad reputations.  A Ph.D. from a university with a bad reputation will actually be worse for your career than having no Ph.D. at all, so you should be very careful.  Plus these programs are as far as I know never free.  You can get financial support for Ph.D. work in several ways (competitive fellowships, a faculty member's research grant, teaching classes for the department), so depending on the field you may not need to pay anything yourself and may even earn a modest stipend, but that's not because the program itself is cheap or free.  Ph.D. supervision is intrinsically time-consuming and expensive.

I'm skeptical of how serious your question is.  Hoping for a free online Ph.D. isn't really reasonable; furthermore, if you have no time to attend a university, then you probably have no time to complete the years of full-time research required to write a Ph.D. dissertation.

However, I felt it was worth answering to warn you about scams from unaccredited universities.  There are plenty of diploma mills that will offer you a relatively cheap and easy Ph.D. online.  Such degrees are completely worthless, so they'd basically be stealing the fees from you.