After giving this some thought, I've come up with a few ideas that would be easy to incorporate into any story, historical or not.
Great characters have lots of layers. Lady Mary is a prime example. Every now and then we get a peek at the great insecurity she feels, which is often made up for in careless arrogance. Interesting combo.
A character's outside appearance hints at their insides. O'Brien looks like a shrew on the outside and guess what...she is.
Characters that aren't overly serious all the time, such as Mr. Carson, make them three-dimensional--and wholly relatable.
I know. Seems like you'd want your hero to be all that and a bag of chips, but guess what? Those are the kind of characters we usually want to slap. Matthew Crawley is a great guy, but he's a little too slow to take charge in some situations.
Memorable characters are surprising. I never know what's going to come out of Violet Crawley's mouth. Oh, I like to think I know, but often it's not what I expect.
A hidden past is a great idea. But don't tell it all at once. Toss out tidbits every now and then. Hint at it, even. Who honestly didn't wonder about Mr. Bates' past?
A compelling character often has a cause which they are passionate about, usually one that involves justice. Lady Sybil Crawley cares about politics, women's rights specifically, which pretty much endears her to every female on the planet.
That seems like quite a laundry list, and I'm not advocating a spreadsheet to check off each trait for each character. Good news: these qualities can be summed up.
The bottom line is a great character has to be relatable... or the reader is going to shut the book and never return. And that's what Downton Abbey has going for it. At times, everyone is as despicable as Thomas or sweet as Anna. Consider that when crafting your next set of characters.