There is not a specific setting for this distance, but there are a couple of ways you could do this depending on the look you are trying to achieve.
First, for a categorical axis (like the bars) you can add \n in the name to force a line break.
If you had to do this for a lot of rows, set up a Text expression column and use that in the command.
To get a finer level of control, here is what you can do…
1 – Hide the default x-axis (uncheck Draw x-axis on the right side of the Axis settings).
2 – Add two Extra Axis commands from the command menu.
3 – Set the Position for both to ‘Bottom’.
4 – Set the Drawing style for the first to ‘Only ticks’ and the second to ‘Only labels’.
Then you can use the sider to adjust the location of the labels.
I would recommend placing both commands in a group. Then you can more easily copy and paste the group to other graphs or drag & drop the group to other thumbnails.
We are curious how much space you are looking for, is adding one line enough or do you really want a finer level of control?