Disclaimer: I have never read Pachinko; this answer is based on the quotes in the question, the Guardian interview quoted below, and my knowledge of the English language.
TL;DR: the first sentence can be paraphrased as
History has not recorded the lives of common people even though it should have, but this is of no importance.
And the second bold bit as
their own rulers hadn’t been able to protect them, though the rulers should have.
This answer segues into a bit of interpretation and broader context instead of just analyzing meaning, but I believe that this is required for a full understanding.
I'll start with Lee's explanation of "History failed us, but no matter", from a Guardian interview.
“History has failed us, but no matter” serves as my thesis statement. I believe history has failed almost everybody who is ordinary in the world, not just the Korean-Japanese, who are the subject of Pachinko. I am also arguing that the discipline of history has failed. It is not that historians aren’t doing their jobs but rather that the memory of history has been reconstructed by the elite, because the overwhelming majority of ordinary people rarely leave sufficient primary documents; they do not have others recording their lives in real time.
The phrase “but no matter” is a statement of defiance. It doesn’t matter that history has failed us because ordinary people have persisted anyway. This idea gives me an enormous amount of strength and hope as a writer because I am an ordinary person. Those of us who may be women of color, immigrants, or working class aren’t often meant to be people who write novels about ideas, but no matter.
What has history "failed" to do? It may be helpful to look at the precise definition of "fail" that is used here. I quote from Merriam-Webster, definition 1a (transitive):
to disappoint the expectations or trust of
History, Lee says, has not lived up to the expectations of humanity. It has not done what it should have done. The key part here is the meaning that something that should have happened did not happen, or at least did not happen to an acceptable extent.
History has not recorded the lives of common people.
Lee brings up several reasons why history has failed to properly remember common people. She says that "history has been reconstructed by the elite". Our understanding is necessarily biased by who was able to tell their stories. The "overwhelming majority" of common people are far too busy living to write detailed treatises on their lives.
We can see this in the paragraph fragment which appears after "History has failed us, but no matter" (quoted here for context):
At the turn of the century, an aging fisherman and his wife decided to take in lodgers for extra money. Both were born and raised in the fishing village of Yeongdo--a five-mile-wide islet beside the port city of Busan. In there long marriage, the wife gave birth to three sons, but only Hoonie, the eldest and the weakest one, survived.
The humans which this novel opens with are resolutely common. They’ve lived their lives together, grown old together. The narration mentions the town they grew up in (tiny), the fact that they had three sons but only one survived (characteristic of poor, common people), and defines them by their common occupations (fishing and taking lodgers). All of these details build a case that these are not the kind of people history writes about.
Their lives are no less worthy of being told and remembered than those of more privileged people, yet, history will not properly do so. The first sentence of the novel sets this up to be read as a failure of history.
This failure is compounded by the world events happening at this time, or, as your second quote puts it:
the twentieth century's legacy of colonialism and war that still fuels tensions in North East Asia
Colonialism and war disproportionally hurt common people. Powerful people can use their privilege to shield themselves to a certain extent. While privileged people write the history books, common people are the ones who struggle to survive, creating stories of resilience and hard work and ingenuity in the face of their world changing. Yet, again, this - this, what history should have recorded, something that later generations will care about - this is something that history has failed to record. History has failed the common people.
The narrator doesn’t simply say that history has “failed”, but that history has “failed us”. “Us” is a group that includes the narrator and the reader. In using this pronoun, the narrator asserts a group identity - a kinship of less privileged people. Another possibility for the meaning of “us” is the human race as a whole. History’s failing to record commoners’ stories hurts our collective memory, cheating humanity, failing “us”.
Yet, despite all this, despite all that history has not done when it should, the narrator brushes it aside with “but no matter”. Quoting Collins Dictionary, American English definition 1 for “no matter”:
it is of no importance
Here, the narration declares that history’s failing is of no importance. Even if no one writes history books about common people, that doesn’t affect their lives. They still survive and live and make their own stories even if no one cares to listen. Lee calls this a statement of “defiance”, one of resistance to the privileged history that ignores common stories.
The same usage of “fail” is present in the second quote, reproduced here so you don’t have to scroll at the way up again:
their own rulers had failed them
What is the failure here? What expectations have the rulers disappointed, what have they not done which they should have? These questions are answerable by looking at the surrounding context.
The men yearn for “another nation to be strong”. Being insufficiently strong, then, is how the rulers have failed. A specific example of this failure is provided
Korea had been colonized for twenty-two years already. The younger two had never lived in a Korea that wasn't ruled by Japan.
“[T]heir own rulers” are the original Korean rulers, who did not live up to the expectation of preventing colonization. (It doesn't make sense, in context, for "their own rulers" to refer to Japanese rulers, because the Japanese rulers have not failed, they have succeeded.) The men believe that if their rulers had been stronger and more powerful, Korea would not have been colonized.
It is significant that “own” is used here, instead of (for example) “prior” or “original”. By claiming the previous rulers as their “own”, the failure is compounded. Part of the social contract between rulers and their populace is that the rulers will protect the people to the best of their ability. If a random adult lets a random child they were watching be kidnapped, that is a failure and a dereliction of the general duty to keep those under one's care safe. However, if a parent lets their child be kidnapped, that is seen as an even worse failure, because parents have a special obligation to protect their children.
The ability of Japan to colonize Korea is a failure of the Korean rulers. The Korean rulers disappointd expectations by not being sufficiently strong to protect their people. Then, not recording the experiences of common people under colonialism and war is a failure of history. History has disappointed expectations by not sufficiently remembering the important stories of common people.
