I work in construction too and metric is how my brain functions, but we’ll probably never get rid of imperial here as we still have old doors in feet and inches that need replaced.
In SketchUp Free, I am able to type dimensions in any of these formats:
3’6
3’6"
2’3 7\8
2’3 7\8"
and they both work in Chrome whether smart quotes are on or off. Safari doesn’t like the " part so it’s better to leave that out, but 3’6 still works. Leaving it out saves typing time anyway.
None of these problems are affected by smart quotes being on or off. If you want to try it for yourself, you’ll find the setting in System Preferences / Keyboard / Text / untick “Use smart quotes and dashes”, but it made no difference for me.
On Safari with smart quotes turned on or off, the " doesn’t work but you don’t have to use it, use the 3’6 format instead of 3’6". I think SketchUp Free works better in Chrome anyway, so use Chrome.
I’m also on High Sierra here, version 10.13.4 (17E199), Chrome Version 66.0.3359.139, Safari Version 11.1 (13605.1.33.1.2) so I don’t know why it’s not working for you.
I can type archaic sizes like 2’3 3/4" in SketchUp Pro 2018 and it works fine, but not as nice as metric.