How difficult is it to obtain a postgraduate qualification in the academic field that's unrelated to your undergraduate programme?
This really depends on what kind of U-turn you want to make and what kind of additional resources you avail yourself of. In this day and age, because we have access to so many resources (from university extension courses to online classes to self study via Youtube and the WWW), a dedicated student can build their skills in a field not necessarily related to their undergraduate degree. 

That said, it depends what kind of U-turn you want to make. To pick up your example, if you have a BA in Art History and you want to get an MS in Physics, I doubt that any serious program is going to let you waltz right in. However, if you took a series of courses through a university extension program, or, perhaps, even online, that covered the terrain of an undergraduate physics program -- and if you also demonstrated your competence and interest in the field through an independent research program, you would probably be able to find a program somewhere.

On the other hand, if you have a BS is a technical field and you want to get an MA, programs are usually more accommodating. In my field (politics and policy), students with an undergraduate BS can often add substantial value because they have an in-depth understanding of math and programming that allows them to gain more traction when analyzing research questions using quantitative methods.

In either case, it will greatly improve your chances of admission if you can articulate why you want to make the U-turn and also explain how your particular path adds value to the program you are applying to. If you find a school and a program that excites you, try contacting some of the professors at the program and see if you can meet or talk with them. During the conversation, you can make the case why you want to move into their field and ask them how you can best prepare yourself.

"This above all; to thine own self be true" -- W. Shakespeare