I guess I can't answer for every field of history, but in my field, yes, it's definitely possible, and something that is becoming more common lately, for the reason that you mentioned - not that PhDs are useless exactly, but there are a glut of people with PhDs and an extremely limited number of teaching positions.

I had a postdoc research job for a couple of years, but I never had a teaching job, and I haven't worked in academia for almost 10 years now. Now I work from home as a translator. I've got the free time to research and write, but I know that's not be everyone's experience. Some people don't have jobs with free time like that, or they just don't want to publish while working outside of a university. For some people the teaching is more important than the writing, so whether or not they could still publish doesn't really matter. For me, I've published a journal article, several chapters in edited volumes, and a couple of book reviews, and I've attended and presented at a few conferences. I've got a few other things I'm working on now, too. At least for medieval history, it doesn't seem to matter at all if you're affiliated with a university or not.

Unaffiliated PhDs like me have various names - "independent scholar" and "alt-academic" are two I hear often. How can we find jobs outside of teaching positions in universities? Should universities help us prepare for non-academic careers while we're still students? Should we be encouraged to keep researching and publishing, or does being outside the academic system mean we're cut off from the scholarly world? What kind of access to books and databases and resources should we have? What about PhDs who are still managing to find teaching positions, but aren't able to get a permanent position/tenure? These are some of the questions that are floating around recently.

I'm lucky to have limited access to my undergraduate library with an alumni card. The biggest obstacle there is that I can't use inter-library loan or access any online databases or e-books. People with university jobs might get their departments to pay for travel to a conference, accreditation to visit an archive, etc., so that's another thing we don't have access to.

Scholarly associations have been trying to help us out lately too, instead of just abandoning us completely, as would have happened in previous generations. A university department might pay for membership in an association, but we have to pay out of pocket for that, so at least there's usually a reduced price for unaffiliated people these days. For medievalists there's the Medieval Academy of America (MAA), the Canadian Society of Medievalists, the Society for the Study of the Crusades and the Latin East, etc.

Networking is also really helpful. I can use social media to keep in touch with people I went to grad school with (whether they're tenure-track professors, independents like me, or out of academia entirely) and people I've met at conferences, or even people I've never actually met in person but only know through Twitter or Facebook. There are medievalist groups on Facebook where people share job postings, calls for conference papers, or resources for books/articles. Twitter is a bit less organized but there's definitely an "[X] Twitter" for all kinds of things (History Twitter and Medieval Twitter in this case). Everyone follows each other so we can see what everyone's working on, and we can share resources that way too.

Within the MAA I know there is some work being done - there's a Professional Development Committee, mentoring programs, and the idea of "MedievALLists" (i.e. everyone's a medievalist, not just the ones with university jobs). There's been some discussion about setting up funds so unaffiliated people can travel to conferences, get fuller access to university libraries, visit manuscript archives, etc.

But these concepts are really quite new, really only in the past few years, so nothing has really happened yet and I'm not sure anything has been written about it. There aren't really any specific sources I can point to, like I would try to do for a usual AH answer. I don't even know if anything like this is happening in non-medieval fields of history.

So, the short answer is, maybe it depends on your specific field, but for medieval history the question of independent/alternative/unaffiliated scholars is an important one, and we're certainly encouraged to research and publish if we want to. I've definitely managed to do it and I know lots of other people have as well. It's a challenge to get access to money and resources, but if you've got the time and the ability, there's nothing stopping you.

Independent Scholar's Handbook: How to Turn Your Interest in Any Subject into Expertise was recommended on another forum.

I purchased a copy and found that while some of the information was dated (card catalogs, limited mention of the internet), other points were very helpful.