Can you export a layer as a obj?
I'm not sure why you want to do this, so I don't know if my answer is relevant, but here goes.
save your file
make sure everything in every layer is deselected
Display only the layer you want to export
Select everything in it
With Shift held down click on all the other layers to display them too.
Invert the selection, with Control I, and press delete
Export the file normally. It only has the objects in that layer left.
To answer your question in the comments - Select Only option is found in the left column of the page you're on after you click on Export as Wavefront Obj file.