I'm using KDE with the Breeze Dark theme, and when I open a Sheet, it isn't very viewable because the cells are a dark-grey, while the font colour is black.

How do I fix this? Can I change the font colour? I couldn't find any setting for that.

You can select the unreadable cells and open the cell format dialogue using the context menu or the format menu. There you have different tabs for font and background that allow you to change the colors of those.

If you do not want to change the formatting of the document itself, you might consider launching sheets with a different color scheme. To do so, you can adapt the file ~/.config/calligrasheetsrc . To get an idea how to change this file to affect the color scheme you might have a look at ~/.config/kdeglobals .
You can set your global color scheme to a bright one, copy all the sections from kdeglobals starting with "[Colors:" to calligrasheetsrc and change your global color scheme back to Breeze Dark. Then calligra sheets will have a bright color scheme with readable cell text by default.

Go to the ~/.config/kdeglobals and add these lines

[Colors:View]
BackgroundNormal=94, 104, 109

the #s representing the rbg value of whatever color you choose.