Honestly, it will be FAR cheaper to just buy a 3.5" IDE/ATA external enclosure, remove the drive, put it in the enclosure and use it that way. Even a top of the line enclosure will be no more than about $50, a cheap one likely under $20 - any place like Apple is going to charge your for their people's time, which will add up to far more.
You can find enclosures at www.macsales.com, tigerdirect, newegg, amazon... It is not hard to remove the drive and put it in the enclosure.
Alternatively, if you have access to another Mac with an empty drive bay (and the Mac is an onlder one, with an IDE interface so an older G4 Mac) you could put the drive in that machine and access it that way.
Its your choice - you can always call a few places and get some estimates to do it all for you (just be sure to tell them this is a Mac with an older IDE/ATA drive so they will know they can handle the odler drive interface AND the fact it will he an HFS+ formatted drive - so a windows-only shop will be useless to you). If you have an reliable computer shop nearby that knows about Apple machines, you could also ask them about actually just removing the drive and installing it in an enclosure for you - you would actually take care of copying the files and such, but just get them to help pull the drive and set it up in a suitable enclosure for you to use as an external drive (heck, offer them the G4 for parts in exchange for the service, if they sell or repair older Mac's?).