Making an image with a rounded rectangle, when I save for web, the image is greatly oversized due to the rounding.
How to get rid of this oversize?
Could you explain more? That logic doesn’t make sense to me. ‘Oversized due to the rounding’? What makes you conclude that? In my experience, the image will save at whatever size the document is saved at… Perhaps describe the basic steps you followed so we can see if it happens to us.
Sorry, apparantly it was due to the Bas Relief effect. I you create one I think you can see what I mean.
(image is taken down to this alone, causing the oversize)
It’s saved as .ai so I cannot upload it.
Hmm, weird. I just drew a rounded rectangle and applied bas relief to it, then saved for web. No problem—it saved at the normal size. 
That is weird. Could I save as a .pdf and upload?
Probably. You can upload pdfs, by the look of it. PDF and .ai are closely related anyway.
Okay, and all actions are removed.
The rounded rectangle is way outside the artboard. The white rectangle is the actual document you are using. Drag the rounded rectangle into the white area, then save for web.
If you go to save for web, it still does it, regardless of how tight the document is set.
It extends from the image itself.
This seems to be the problem - the raster settings. Going to;
Effect > Document Raster Effect Settings
Current document is set to add 36 px around object.
However, when as you reduce this to reduce the surrounding image, it does change the bas relief effect.