Your best bet is going to be an older Mac than a Power Mac. The easiest solution would be to find another Classic, or to find an LC or LCII, as both could run System 6.0.7 or 6.0.8 (very commonly used/shipped with the original Classic). Other good choices are the Mac II series or the Mac SE (with a high density drive, sometimes referred to as a SuperDrive or FDHD drive). I wouldn't recommend using a Quadra, Centris, Power Mac, Classic II, Performa, or LCIII/higher numbered LC. PowerBooks use smaller drives and would be unable to accommodate the drive from the Classic, most Macs made from late 1994 onward use IDE, and the Plus and earlier can't accommodate internal hard drives. Stay away from original IIs and SEs without SuperDrives, since they can't use the 1.4MB disks you'd need to transfer the files to a USB floppy-equipped Mac. Don't forget to find a compatible monitor if you're using a Mac without an internal display.
The easiest way to do this, which is why you'd want an older Mac that can run System 6 (newer ones require System 7), is to transplant your hard drive into the case of the Mac you've chosen for the transplant.
Another good choice is to use an external hard drive enclosure with a SCSI connection, although these aren't all that common these days. You'll find plenty of Classics, SEs, and LCs out there (other models are a little less common; these seem to show up all the time since they were used extensively in schools).
Assuming you wish to transplant your drive into another Mac (the easiest way since most Macs, II series excepted, typically only accommodate one drive), you'll need to remove it from your Classic, remove any existing drive from the new Mac, and place the drive in the new Mac. You can then use a floppy (stick with a 1.4MB to ensure compatibility with USB drives).
The trickiest part is going to be removing the drive. You'll need a long-shafted Torx T15 driver. There are two screws in the Mac's handle and two near the ports. Remove these screws and gently pull the two parts of the case apart. After you open your Mac, you'll see the hard drive sitting on top of the floppy drive. There are two Phillips head screws holding the hard drive bracket on the floppy drive bracket. Remove these, disconnect the power and drive connectors from the back of the drive, and gently slide the drive bracket out. You may need to loosen a cable from the right side of this bracket.
Next, you'll need to prepare your new Mac for the drive. SEs open identically to Classics and have their hard drives mounted the same way. LCs are easy--the case snaps off (some put a screw in to deter this, but there's only one you'd need to remove on the back above the ports if it did exist). Remove the old drive, connect the new drive, and you're in business.
The only problem you may encounter is if you are using a machine other than another Classic and you once installed a system specifically for use on a Classic on your hard drive. In that case, your computer will state that it cannot boot. There's a very easy fix to this--either install a system from your original system disks or use Apple's disks (available on their website under the Apple Archive section of old software) if you lost yours. Install it for "any Mac" or for the model of Mac you've got.
If you need a Mac, please feel free to post in our trading post section and hopefully someone will be able to help you find one.