IANA real estate professional, but 'm not unaware of any jurisdiction where an agent is required (though I’m sure many of them absolutely love the idea that people seem to think so).
(Note that “Realtor” is a sort of real estate agent “brand” applicable to members of the National Association of Realtors. They probably like the idea that their brand is well known, but not the idea that it gets used as the generic term.)
Yes, it is legal. People having been doing “for sale by owner” sales long before the internet existed. The mechanics aren’t any different then having a real estate agent, except that you would have to do all the legwork, including hiring a real estate attorney to make sure everything is in order regarding the title, mortgage, etc, at the closing.
One for sale by owner website, fsbo.com has an “Education” section that goes more into detail.
You can (again absent some strange local rules I’m unaware of) advertise real estate as you wish, including on Craigslist, by putting a hand-lettered sign on your lawn, etc.
For it to legally and properly change hands, various paperwork is required. The creation and filing of this ranges from reasonably straightforward (in the majority of cases) to seriously complicated.
Any lawyer who claims to know how to do this probably does (it’s common, and definitely not rocket science). I’ve done this smoothly without the involvement of a lawyer, but that is not likely to be a good plan for most people most of the time.
A bank (or similar financial institution) certainly has to be involved in a mortgage, since that’s where the money being borrowed comes from.