I don't think it can even be called a grid, as the gaps between points aren't the same length.

depends on your choice of floating point standard, you're right for the current IEEE one

This is a classic example of something that should be fixed point, not floating point. Just represent positions internally as an integer between 0 and some large maximum. You could use the number of screen pixels, or a larger value if you want to allow e.g. precise moves by zooming in.

If the gaps are all the same size, isn't it fixed point rather than floating point?

I mean, if your board has a fixed zoom level you have to fix a certain precision

are you able to represent it some place else?

what does this mean?

You stared real numbers are not representable, but specified “on computers”. Is there some medium they can be represented upon? Math papers is my only guess - but I read those on computers :)

*stated