This is a really good question and has made me think twice (or lots) about the issue. I’ll be the first to admit that I had no idea whether strawberries (and plants in general) could get cancer before you asked. But it appears that they can.
A cancer (or tumour) is basically a region of uncontrolled growth in an organism. This occurs because of some problematic issue with cell division control, i.e. for some reason the cells in question start dividing uncontrollably without the usual stop signals. This leads to a lump of cells being formed that would be recognised as a tumour. We all know about all this happening in humans (and animals, I am sure you have had or heard about pets who dies of cancer) but I wasn’t sure if it happens in plant cells. My gut feeling was yes.
This turns out to be the case and sometimes infection of plant cells with a bacterium or a virus will lead to the scrambling of the genetic information contained in the DNA, meaning that the cells start growing in an uncontrolled fashion. This leads to the formation of an amorphous lump (also called gall) unlike any of the plant’s recognisable organs, which simply drains energy from the plant and does not have any constructive function. So when you next see a spot of imperfection on a plant leaf, that may be a tumour… I can’t find any references to strawberries in particular but hopefully I’ve answered with plants in general in mind at least!