There most certainly are, at the information density of DNA. DNA is about a thousand times more information dense than our best hard drives. At that level, a passing photon can change or break enough chemical bonds to alter the bits. A nearby high-energy molecule can do the same. This actually happens gazillions of times per day in our body and we have a few hundred enzymes to automatically correct errors and a few hundred more to detect when those can't fix the problem and shut down the cell. The fact that it works at all is jaw-droppingly amazing. We take copying bits on and off hard drives for granted, and they only work for a few years max--cells operate on an entirely different scale of information transfer and yet some multicellular organisms live for centuries.
Unfortunately, I don't think you could improve the error checking mechanisms in our cells without fundamentally redesigning a lot of how we work. It would be interesting to try to add more "parity" mechanisms besides the complementary base pairing, which is used by most of the repair enzymes. From an evolutionary standpoint, no species would ever naturally develop perfect DNA replication because it would halt diversification, which is needed to survive continually changing environmental threats.
I should also add that it's a misconception that every cell in our body has the exact same DNA. That's only a half-decent approximation. In reality, a lot of our cells accumulate trivial mutations and it's OK. Some cells even rely on editing their own DNA to perform their primary function: that includes the B and T cells [1] which the cancer in the OP arise from.