You can find "short depth" rack mount server cases (chassis) that aren't heavy. Newegg offer them for cheap.
With a 4U size case, you'll very likely be limited to one or a pair of 80mm rear fans (which have to run faster and thus are louder) instead of 120mm (which can move more air when spinning at a lower speed and thus can be quieter), although some cases also mount a 120mm fan (or 2) in front for pulling air in.
Ideally, you'd want larger and/or faster fans at the rear pulling air out of the case, and in a rack you might get away with a pair of 80mm rear fans without noticing any noise from the front. OTOH, a 5U case will cost more but will typically mount a 120mm fan (or even 2) in back.
Swapping in quieter fans isn't usually very difficult or very expensive. Mounting the fans using rubber isolators, using rubber grommets on hard drive screws and even a rubber gasket on the power supply mounting surface can help reduce vibration. Lining the case (or at least the lid) with sound deadening foam mat can cost a bit but can be worth it. Plenty of DIY case quieting info available on sites like silentpcreview and htpc sites if you search for it.