For example, is anything happening in the world today a direct and continued response of, say, World War I or II? The American Revolutionary War was over 200 years ago... at what point did the historical context become "solidified", such that when discussing politics or current events, one is only discussing America as it exists now, and not also addressing Revolutionary- or pre-Revolutionary-era battles and treaties and such?
Obviously Islam is quite old, but how much Islamic history should I look into to get an understanding of today's Islam?
It seems like we are entering a new cold war with Russia; how much does the history of the first cold war play into my understanding of current US-Russia relations (whether or not we really are going into a new cold war is up for debate, I suppose)?
Those are all just examples, I don't necessarily need answers to those specific questions. I am not really sure how to phrase the question I am asking. Perhaps: at what point in time does the current "story arc" of our world begin?
Well, in order to understand what is going on in the world today, I think most people would agree that one needs to understand the development of Europe and the Middle East.
To understand that, one needs to understand (amongst other things) the development of Christianity (1st century AD) and the development of Islam (7th century AD). To understand that, one needs to understand the development of Judaism (much earlier) and the development of ancient Greece.
To understand the development of ancient Greece, one needs to understand their main enemies, the Persians.
To understand the Persians, one needs to understand the other Middle Eastern and Egyptian Empires which preceded them.
To understand these earlier Empires, one needs to understand the transition from hunter/gatherer societies to agricultural societies.
This will also be a helpful starting point when trying to understand the histories of other parts of the world, such as China, India, the Americas, and Africa, which are of increasing importance in understanding what is going on in the world today. (Not to imply that the histories of these other parts of the world are not more and more important to understanding what is going on in the world today, but the history of Europe and the Middle East is probably more important, particularly given how much Europe and Islam have influenced the rest of the world in the last 500 years.)
In short, I think if you go back to the initial transitions, roughly 10,000 years ago, to agriculture, you will have enough historical context to understand most of what is going on in the world today. (Though you will be missing out on some context from what happened before agriculture was developed.)
I don't think there's really one solid number of years I could give you - the greater and more holistic understanding of history you have, the better informed you will be.
To understand Russia you need to not only understand how its current system and leadership developed throughout the Cold War and the Soviet Union at large, you need to also know about what came before it - Tsarist Russia, the Russia of the Peasants that inspired the Soviet Union.
To understand the Middle East of today, your most direct need requires you to go back to the dismemberment of the Ottoman Empire, and how the 'Mandates' came to be, but that really only explains the political situation and the region's arbitrary borders. But that doesn't even begin to give you proper context for Islam, or the role the Iranian people have to play as a traditionally dominant party in the region and the seat of many Imperial regimes.
No history exists in a vacuum, everything is built upon something that came before, and to understand one you need to understand its predecessor otherwise you're lacking context.