I'm building a chest of drawers out of reclaimed barn wood. For the drawers I've used 1/2" sanded ply for the sides and 1/4" for the bottoms. They're pretty darn smooth but I'd like to put a clear coat on them to protect them. I've read that poly can cause undesirable odors so I guess I should avoid that. What about water based poly or shellac? Seems it's pretty much 50/50 on whether people even finish them at all (vs. unfinished/unprotected) but I'm looking for some practical advice on the topic.
Major re-write as I missed that you were asking specifically about the drawers themselves.
Seems it's pretty much 50/50 on whether people even finish them at all
The traditional practice was for drawer bodies, and in fact the entire interior of a chest of drawers, not to be finished in any way. There are possibly multiple reasons for this, one of which might simply have been penny-pinching!
But one reason that is mentioned in some older books is that the contents of the drawers might stick to any finish in warm and/or damp weather. This is not likely with any finish that dries hard however, and particularly not if a very thin coating is applied (i.e. not enough finish to form a distinct film on the surface of the wood).
I've read that poly can cause undesirable odors
If you wanted to use oil-based poly you could wait for a full cure before inserting the drawers. One of the indicators of a full cure is that the finish has no odour with your nose right at the surface. This might take as long as a month though!
What about water based poly or shellac?
I think shellac would be the better choice between these two because it's so easy to make a very dilute version, which would provide just that bit of protection that you might be looking for.
An advantage of using oil-based poly over waterbased poly is that the former can easily be thinned to a "wiping consistency" (note: can still be applied by brush) just by adding additional spirits, while with waterbased poly apparently you shouldn't thin them heavily with water as it interferes with how they dry and harden.
Waterbased varnishes have one other potential disadvantage worth nothing, as with all finishes that are based on water there is the possibility that the first coat of waterbased poly will "raise the grain", that is, swell the surface wood fibres and make your previously super-smooth sanded surface rough again. There are a number of ways of dealing with this but probably the best is to deliberately raise the grain in advance by wiping over with water. After the wood has dried you lightly sand to remove the raised grain. Some users do this two and even three times to ensure that when their finish goes on the wood it will remain smooth.
I wouldn't be afraid to using straight polyurethane. I build shelving for my room in the form of a large book case and a small dvd shelf. I covered both in straight poly and after drying it neither has a smell or any stickiness to it as graphus suggested it possibly could.
The only issue I had with it was that I only used one coat and didn't sand. I found out later that it stood the grain up slightly in some places. Not enough to really notice, even when running your had over it, but I sure notice it when I try to dust and dust and the duster try to stick to it. So if you do a couple of coats and sand lightly between, I think you'd be good to go. It will also help protect your wood pretty well. There is a slight shininess to my boards, but I think it looks good and I'm sure a buffing with a very high grit sandpaper would nock the shine off. If you were looking for something that wasn't as heavy, I've think you can mix water with the ploy to make a thinner wiping varnish, but I think that's mostly an issue of how easy it is to apply and how tough it is. I'm sure you have similar issues and benefits to full strength poly, just to a lesser degree.