In Capture One Pro 8.x Is there a way to strongly blur the background of an image? I tried using an adjustment layer and the negative slider values for "clarity", "structure" and "sharpness" but that doesn't blur the background strong enough. I would like to blur it in a way as if it was out of focus (shallow depth of field).
Not directly, no. All of the control you have within Capture One Pro can be summed up as emphasizing or de-emphasizing the existing sharpness in the image. That is the normal state of affairs with raw image processors. You would need to edit in an external application to actually blur the image (or selected parts of it). If you aren't using a separate image editor at the moment (Photoshop or similar) that has synthetic lens blur, there are plugins that can be used stand-alone (usually with a minimal "host" utility) such as Topaz Lens Effects or Alien Skin Bokeh that will let you give the background a believable out-of-focus blur.