/diy/ - Do It Yourself

I'm taking an intro machining class in university. We're doing basic shit on the school's lathes and mills. I was given a couple part diagrams with with various tolerances in the +/-0.001-0.003" range.
How do I hit this with shitty measuring tools and sloppy machines? All the calipers and micrometers in the lab are beat to shit and are frequently a few tenths off from each other. The machines are fucked, like the lathe's have visible runout and the hand wheels have to be spun like a half turn before the tool will start feeding, and even then the feed is really more of a suggestion than anything reliable.