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I have made this simple shape from two planes, it has no volume.

2D shape in front ortho

As you can see there are two faces that are overlapping in the center creating that weird square, which is not a face and cannot be deleted or dealt with.

One of the faces selected in the edit mode

So the question is, how can I remove that thing in the middle and make sure I'm dealing with just one face?(I've tried pressing F to combine two faces but nothing happens).

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    $\begingroup$ Don't think you can use Boolean for this. One option is to go with Knife Project from the top view (but planes have to be separated into different objects), another is using tinyCAD addon. $\endgroup$ – Mr Zak Sep 11 '17 at 21:29
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    $\begingroup$ Possible duplicate of How can I add vertices to intersection of two edges? $\endgroup$ – Mr Zak Sep 11 '17 at 21:51
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For such task there are Mesh > Faces > Intersect (Boolean) and Intersect (Knife) tools in Edit Mode, but for your example they are not working as supposed to. Also with two separated shapes from your example Boolean Modifier (Union) isn't working as well. I can't say why, as I'm not using Booleans at all.

As a workaround you can make it using Knife Project tool.

  1. Select one of your shape/face in Edit Mode then use P > Selection.

enter image description here

  1. In Object Mode, select one object, then with Shift select second one and go to Edit Mode.
  2. In the Tool Shelf choose Knife Project.
  3. Repeat steps 2 and 3 but now select objects in different order.
  4. Join both objects Ctrl+J.
  5. In Edit Mode select all then W > Remove Doubles and use X > Limited Dissolve.

enter image description here

Note: Consider using quad topology as this will save you some problems later on. Working with ngons could be problematic.

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You can use the Boolean 2D Union add-on for this.

Select your mesh and press W. You will see new options in the Specials menu.

Choose Boolean 2D Union to merge the separate faces into one connected piece of geometry.

Boolean 2D Union

This is the result:

Boolean 2D Union result

Similarly, you can use the TinyCAD add-on's XALL function to do this with edges that don't have faces. (This add-on is bundled with Blender.)

TinyCAD XALL

TinyCAD XALL result (notice the new vertex in the middle):

TinyCAD XALL result

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