You can get your own width by putting an empty div somewhere in your component:
<div aura:id="measurement" />
var width = component.find("measurement").getBoundingClientRect().width;
Note that this won't work in "init" because rendering hasn't occurred yet, but you can do so immediately afterwards using setTimeout:
var width = component.find("space").getElement().getBoundingClientRect().width;
The SecureComponent API has limits, but as long as you stay with normal HTML or your own custom components, you can get this measurement.