What I've read before is, if the 2 drives are of the exact same model and make (or if you have other reason to believe this procedure is feasible), you can transfer the disk platters from the failed drive to the working drive. If all goes well, the new drive will read the old platters, and you can recover your data off the drive (assuming the platters are intact).
Mind you, while it's simple enough in concept to be described in one sentence, in reality it involves finding a dust-free environment, VERY careful disassembly of said drive and handling of platters (get one fingerprint or significant speck of dirt on those platters, it's game over unless you're lucky). In other words, not something you want to try if it's your first time.
You definitely want to do lots of reading up, and practise on an old/cheap drive or something. You will likely also need specialised tools; platters are not easy to remove without contamination (try disassembling an old drive).
Some details can be found
here.