Not really. Everything you’re studying is long gone, so it’s not like there are going to be any sudden “fossil emergencies.”
There are the usual job stresses, of course. You do need to cross your T’s and dot your I’s when it comes to writing an academic paper or book. For example, Bill O’ Reilly came under fire for his book, Killing Lincoln, when historians pointed out its historical inaccuracies. O’ Reilly refers to the Oval Office in his book, even though the Oval Office wasn’t built until 1909. He also stated that Ford’s Theatre burned down in 1863, when it really burned down in 1862. What a difference a year makes. You can find yourself in a world of hurt and ridicule if you mess up any facts.
Also, historians who teach have busy schedules. A typical day for a history professor may include developing a lesson plan, grading papers, being available during office hours, performing research for a book and dealing with editors. That being said, historians do have pretty flexible jobs that allow them some family time.
Honestly, the biggest stressor you have to deal with is the possibility of being fired from your cushy university or museum job, because if you can’t find something else within your field, you really have no portable skills. Clean-up in aisle five.