Hi Lars,
Tables in Numbers can have gridlines (Table properties) and Borders (Cell Properties).
You can control gridlines by clicking anywhere in a table and Format Panel > Table > Gridlines

Click on each blue button to remove those gridlines. (This is also where you can change or remove the default Alternating Row Colour).

Now for the Cell Borders:
Select the whole Table by clicking on the "bullseye" top left.
Format Panel > Cell > Border
Choose All Borders (the icon top right under Border). Choose No Border.

This is also where you can get rid of Cell Fill.

Choose No Fill.

Gone!
And gone in Print to PDF 😉.

Regards,
Ian.