Recently tenured (last 3-4 years) professor of history here. In short, everyone else has hit most of the points, but I'll add data.
No. Unless you're already independently wealthy, it's unlikely. Some fellow grads did sign up as couriers to get the occasional quick trip to Kuala Lumpur or something but that hardly qualifies. If you're lucky and in a really good program, your stipend may cover your research expenses. If you study certain subjects (African history, I can tell you, but a lot of overseas history work) it won't cover enough.
If you get into a strong program, you may get a year of research fellowship support. That's when you go. In my work, I spent that year in my African focus area, and I spent some summers in the UK, France, and Germany working in supporting archives. Many programs, however, including the one where I now teach, do not offer ANY support unless you are teaching or serving as a research assistant in the same academic semester. I went to a top-10 program for my field, and I got 3 years of fellowship out of 6 years of support. If you don't get those non-teaching years, you will not be able to complete and be competitive unless you incur major debt.
Time/money for general leisure? My normal reddit reflex is to type "HAHA" over and over and over again, so you get the point. As the years went on I did have more time for such activities, and I was lucky enough to land a few one-year visiting posts while ABD that paid really very well. I still barely made ends meet. I can't state this enough: if history is not your overriding passion, if it is not what drives you, if your travel for research isn't basically your idea of 'fun' (or a large part of it), you're gonna have a bad time. If you have other hobbies and interests it will require a great deal of effort to carve out specific time for those. You should, of course, take down-time when you can, but it's not easy and you must be thrifty.
I'm tenured at a state flagship, AAU Research-1 university in the US, which is the position you're describing. According to the AHA and the AAUP, I am atop the second quartile for my rank (paid better than about 75% of my peers). Yes, I live relatively comfortably because my 'life costs' have not risen from the days of a pauper's $16,000 stipend. Now that I make the princely salary of about $73,000, I do pretty well for myself, although much is spent catching up with lost retirement savings time and milestones that old college friends passed a decade (or more) ago. Is my life leisurely? Hell no. I work a lot, to stay atop my field--it's how I got the job, how I sailed through tenure, and the only way that I obtain any increase in pay (unless I get an offer elsewhere). There's more time to relax now without the tenure clock, but the pressure was relentless thanks to the flooded job market, and it is a constant battle for work-life balance.
Beyond that, in terms of actually following the path, you must already know that the raw odds are against you. Not knowing where you got your BA, what your actual area of interest is, and how well connected or prepared you are, I can't really say what you chances are like of getting into a top-10 or top-15 program that might afford you some degree of the balance you seek--but be aware the demands of your path will push against you.