I would ask for the hard drive back, and put it in one of these:
[produto vinculado ausente ou desativado: IF107-097-1]
You can get a USB2 version even cheaper on eBay, in the $15 shipped range.
With the drive in an external enclosure, you can browse through the files on it and salvage whatever you want yourself.
$300 would tend to be somewhat of a ripoff, but I'd get some clarification as far as what they would be doing, and what exactly they think "gone bad" means. Often an operating system will become corrupted and will no longer boot, in which case the drive is perfectly readable in an external enclosure, and this is the ideal situation. But if the drive itself has a defect and is unreliable, the tech may be intending to use elaborate recovery software to re-construct your files, and this fairly advanced procedure could account for the $300 cost. However, true high-end disk recovery facilities usually charge $800-$1200 per drive, so the $300 figure makes me wonder if this is a hack who doesn't necessarily know what he's doing, or if your files are easily recoverable, and he's just trying to make easy money.
Either way, it can't hurt you to put the drive in an external enclosure, because it will let you try to salvage the files yourself, and if the drive ends up making odd noises, does not appear on the desktop, or does not let you copy files over, at least you have verification that the drive has problem. At that point you can always pay someone to assist you, and having the drive in an enclosure will make it easier for them to access as well.