How to create an outlined circle with centered horizontal and vertical gaps in Illustrator?

Create the circle shape.

Remove the fill.

Give it a stroke that is as thick as the width you want.

Go to Object → Expand.. in the top horizontal menu.

Create a rectangle that's as thick as you want the gaps to be.

While you still have the rectangle selected, hold Shift and select the circle. Use the alignment options to center it horizontally and vertically relative to the circle shape.

Select only the rectangle. Press CTRL + C to copy the shape.

Press CTRL + Shift + V to paste it in exactly the same place as the first rectangle.

Right click the currently selected shape (rectangle 2) and go to Transform → Rotate... Enter 90° into the dialog box and press enter.

Select both rectangles and go to the pathfinder. Merge them.

Select the newly formed cross and the outlined circle and, using the pathfinder again, select Minus Front.

Et Voila!