There's tons and tons of error checking- we have at least 5 different error correction and repair systems in DNA, cell cycle checkpoints, and extreme redundancy and feedback homeostasis at nearly every level. Every individual cell has it's own 4 copies of almost every critical gene- two of each chromosome made up of two strands of DNA each. Human bodies can function 70+ years, sometimes with no medical care- something no computer or man made complex machine comes close to.

Beyond specific diseases we understand, it's still mostly a total mystery why we aren't immortal- we have not yet identified what is the basic mechanism of aging, or why it happens at different rates in different species, and mostly our systems are fundamentally capable of repairing and regenerating almost anything, but for some reason get worse and worse at doing so over time. Moreover, this doesn't seem to happen in all organisms- there are many animals that live ~4x human lifespans, and at least one species of jellyfish that is biologically immortal.

Redundancy is not error checking. The "error correction" mechanisms are actually just "proofreading" mechanisms and are almost entirely local and centered around transcription. Common mode errors are harder to induce due to the plain redundancy of DNA pairs but also not impossible, and once induced, are impossible to locally notice or correct. In some cases the "error correction" machinery is the cause of these induced errors. The result is genetic disease and/or cancer and is a case of missing error _checking_. Perhaps my definition was exceptionally parsimonious.

> with no medical care [...] something no computer or man made complex machine comes close to.

That's because we get far more units of "work" out of our machines than the person living for 70 years with "no medical care." Some people live just 30 years with no medical care too. And the machine does not need to sleep. We eat food they eat lubrication oil. I don't think this was a good analogy.

> it's still mostly a total mystery why we aren't immortal

While we haven't pinpointed the mechanism, we have a pretty good idea of why, and where in the system we should be looking for the answers.

> but for some reason get worse and worse at doing so over time.

You are a living Ship of Theseus and these "error correction" mechanisms are not perfect. Aside from this there are known genetic disorders which alter the rate at which people age. This is not nearly as mysterious as you're making it out to be.

> there are many animals that live ~4x human lifespans

And what are their resting respiration rates?

> and at least one species of jellyfish that is biologically immortal.

In theory. We haven't found an immortal one yet. They all die. They're also nowhere near our level of biological complexity or capability.