Cancer is not a disease in the same way as, say, hepatitis. Every cancer is different - and not just e.g. skin cancer versus lung cancer, but really even John's lung cancer versus Peter's lung cancer. The only element all the various types of cancer have in common is that some cell in the body basically turns parasitic, and starts dividing uncontrollably, while tapping the body's resources.
Fundamentally, this is a result of the breakdown of the usual role the cell has. This is always a result of DNA damage - of course, the most famous source is from radiation (nuclear instruments, UV light, radioactive carbon and potassium etc.), but the reality is that each cell in your body is in a constant fight against deterioration, most notably from oxidative stress and infection. DNA is quite stable, but even the strongest bond can be broken if you "hit" it forcefully enough. However, there's a limit to how stable DNA can be, because evolution requires some kind of instability to introduce new variations in the genome.
And that's the real rub. You need something unstable enough to allow changes which are (extremely rarely) positive, and give you a better adaptation to your environment, as well as giving your species as a whole a better resistance to change). This is the same thing that gives you cancer.
Okay, you might think, why not make it so that change is possible in sexual cells, but not all the other cells in your body? There's two tricky points. One is that different cells use different portions of the whole code, and in fact, two cells of the same type may have different "effective" DNA, based on which of the two variants (one from mother, one from father) of the code is used in any particular cell. If this was visible, you'd appear striped - in fact, it's what causes striped colouring in calico cats. But the main point is even trickier - in most multi-cellular complex life on Earth, the same mechanism is used for dividing sperm cells as every other cell. Even worse, the same main mechanism is used during usual cell operation, not just during division - all those proteins your cells make are first copied from the DNA, which needs to be partially "unwrapped" (which makes it more vulnerable).
Why not have a different mechanism? The fun part is, there have been different mechanisms in EarthLife's past (and they're still present in many organisms). It's just that DNA is the best we've got - evolution didn't stumble on anything better yet. Our cells are full of mechanisms to prevent cancer, and our bodies as a whole are as well. The problem is that there's so many cells, dividing (and dying) all the time - even with all the repair mechanisms, once in a while, a cell goes "rogue". Almost always, it's very quickly destroyed from the outside (or happens to starve etc.). Only when all the mechanisms fail do you get a tumour, and even then, it's not necessarily the "real bad cancer that kills you eventually" - most growths are not deadly. You need a growth that can "bleed your blood dry" of nutrients, or that interferes with normal functioning of the rest of your body (e.g. blocking blood passage, destroying surrounding tissue).
The same things that enable cancer are really the same things that are required for life to work (and evolve) in the first place. Even if it were physically possible to make a data structure that's impervious to all damage, you need to "damage" it in order for cells to divide, and possibly to read the data (that's the way it works with DNA). If it's hard to do that "damage", you will grow slower, and the changes in the populations will be rarer - in other words, some other thing, faster growing and/or more adaptable, is going to eat you or starve you. No repair mechanism can ever be perfect, because the damage can always occur in the repair mechanism itself. Our own bodies' repair systems are incredibly reliable (it's necessary for any large, long living multi-cellular organism), but they're applied very often. And of course, we're under pretty much constant attack by other organisms that seek to change our body processes for their own benefit (various parasites including viruses).
The best you can do is remove as many causes of damage as possible. For example:
- The aliens live deep in the ocean, and are entirely shielded from most radiation.
- There's no free oxygen (and other strong oxidants) in their environment.
- There's no parasites smaller than your own cells, or parasites that overwrite your own genetic information (e.g. viruses); this is a bit tricky - you'd need something that reliably destroys small things without damaging bigger things.
- The alien's body is resistant to systemic changes and localised damage. For example, cancer is quite common in plants, but it has no way to spread systemically, and damaging one part of the body rarely does significant damage to the whole. This can make some quite alien aliens indeed ;) If you take an ant colony as a single life-form, this is their strategy against cancer as well - cancer doesn't spread between individual ants, so unless it affects the queen, the colony survives.
- The metabolic rates of everything on their world are much lower than ours. This basically gives the repair mechanisms more time to fix any problems before they grow too much. This is basically how chemotherapy works - it prevents all your cells from dividing (which causes the infamous "hair falling out" etc.), which hits the cancerous cells by far the most, since they divide "unrestricted" under normal circumstances.
- There's a systemic repair mechanism ("genetic white cells/antigens") that continually identifies any deviations from "standard" genetic data, and marks the cells for termination. Of course, this would be extremely expensive and might cause you quite a bit of trouble if that's what gets broken - your marker would start marking every healthy cell for termination, ouch. It might be interesting for your story though - for one, it might be something humans might want to adapt to their own physiologies, and two, it might create a "cool disease" in the aliens that could be transferable to humans with disastrous effects. And with a repair mechanism like that, the aliens might lack one of the mechanisms that causes biological ageing - potentially making this whole thing even more attractive to humans, with promises of "immortality".
: Okay, not exactly the same - but they have the same weak point, DNA replication and transcription. DNA is at the center, not the exact replication process - you need information storage strong enough to withstand environmental damage, but weak enough to allow replication and transcription.