Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of every finite-sheeted covering map of a finite graph be embedded in $\mathbb{R}^3$? This is the level of generality I'd be most interested in. On second thought, and after reading the comments, I think a community wiki list of interesting examples (like those below) would be more helpful to other instructors. If you have any more, please post them!
I recently taught my students about covering maps of topological spaces, using the classic example of the real line winding onto to the circle. This covering map can be exhibited in real life (well, not all of it, but a representative chunk). All other covering maps of the circle really can be exhibited with string. Other things that can be exhibited include winding the upper half plane onto the punctured plane, although this is equivalent to the real line covering the circle.
After some thought, it seems that some graph coverings (like a double cover of the figure eight) can be realized in real life, while it seems that nontrivial surface coverings cannot be exhibited unless the circle has boundary. My question is,
What is the class of covering maps of graphs or two-dimensional CW-complexes that can be realized in 3-space, i.e. so that there is a homotopy of the covering space in $\mathbb{R}^3$ onto the base space which is an isotopy for $0\leq t< 1$ and is the covering map for $t=1$?
(This is my working definition of "real life homotopy", since we allow things to touch but not pass through each other. There may be a better definition.)
Best Answer
My favourite example of a real-life covering space is a multi-level parking garage, along with all up-down ramps. Wander around for a while, and you're on a different deck.