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.