Short version: If your device is "much different" from a standard 5" mobile phone, you're out of luck.
The problem here is that a cardboard web app doesn't know what sort of Cardboard viewer it's in. So the best it can do is split the screen in half for left eye & right eye. What you'll find is a developer will "calibrate" the web app (say geometry related to inter-ocular-distance) to the device they are testing with, most likely a 5" phone in a standard Google Cardboard.
Native Cardboard apps have access to the calibration profile for the actual Cardboard viewer the user is using. You've probably seen those QR codes stuck on the side of Cardboard devices... they're important. You'll find native Cardboard apps on a Nexus 10 change the display to suit your Cardboard viewer (although tough to fit into a standard Cardboard!). Unfortunately that profile information is not accessible to cardboard web apps, yet.
I have a similar problem using a Nexus 7 (with Durovis Dive 7 Cardboard). The stereo image fills the display mostly okay, however it's difficult to make the image converge (overlap) and produce depth effect. If you concentrate for long enough, you can almost do it, but you'll end up with a huge headache!
Note: https://www.google.com/get/cardboard/viewerprofilegenerator/ is where you can generate a custom "viewer profile" for your device. But this isn't going to help you with web viewers like the Sketchfab one. I'm sure the guys here are busy working on a native cardboard app to view Sketchfab scenes!
ps. happy to receive new information about this problem.