Not really. Tritium is Hydrogen. It cannot bioaccumulate. Each atom of Tritium will spread out to become one among the quadrillions of atoms of Hydrogen in our body. Most will get out of the body in a matter of days, long before they've had a chance to decay. Even when they decay, they undergo beta decay, which is not very damaging. But even if it were damaging, the damage would be very localized, it would affect at most one cell, and the immune system is easily able to handle that.
To the extend this is true, it implies that it is the square of the radiation dose that determines carcinogenic effects: Half the dose would cause only a fourth of damage.
you could not be more wrong about this. Alpha and Beta are fairly safe OUTSIDE the body. In the case of Alpha it is not able to penetrate the layer of dead skin on your hands.
Internal, it is VERY damaging because there is no "dead skin" to protect the internal organs.
https://www.epa.gov/radiation/radiation-basics
"Some beta particles are capable of penetrating the skin and causing damage such as skin burns. However, as with alpha-emitters, beta-emitters are most hazardous when they are inhaled or swallowed."
Tritium is very different. It is chemically just Hydrogen, which is present everywhere in the body. It just can't bioaccumulate.
In any case, if you don't believe me, here's a statement from the FDA regarding the tritiated water released by Fukushima:
Tritium presents an extremely low human and animal health risk if consumed and any health risk would be further minimized with the dilution effects of discharge into the ocean.
However some water does come out of the reactor you are correct.