My first suggestion is that you try mounting the old HDD in your newest computer and connecting it there. Of course, that assumes that your newest machine has an IDE port on the mobo to work with. If it does not, how about trying another of your slightly older machines that does have an IDE port?
This path is based on my assumption that your external HDD enclosure does not know how to deal with older HDD's with different access systems, but a computer mobo's BIOS will.
In connecting that unit, remember that you'll need to set its jumpers according to how you connect it. If it is the only drive on its IDE cable, connect it to the END connector and set its jumper to be the Master unit on a single-drive setup. For all details of settings, see this link on the WD website:
http://www.wdc.com/en/products/legacy/Legacy.asp?Model=AC2700
If you set it as the Master (END connector) or Slave (MIDDLE connector) on a cable with two drives, set the jumper according to that web page. Ensure also that the other drive on the cable has its jumper set correctly. For this, do NOT use that web page. Use the diagram on the other drive to set its jumpers.
So, once you have that old HDD connected, boot into BIOS Setup. You should see all the drives on the opening page. The BIOS MAY detect that old drive correctly and show you correct identification of its model number and size. If so, exit from there to boot and see if the old HDD shows up properly in My Computer where you can access its data.
If the BIOS does not figure it out, you will need to make manual entries. Set the HDD Type to 47. I think that's the correct type - it is the last possible choice of type number. This allows you to enter your own set of HDD parameters. Then you enter the data from that web page for Cylinders, Heads and Sectors per Track, and possibly for Landing Zone and Compensation. LBA Mode should be ON, Block Mode ON, 32-bit mode ON, and PIO Mode AUTO. With those entered, the BIOS should be able to show you the correct HDD size (730 MB I think) and identification. If you had to enter such data, don't forget to SAVE and EXIT from Setup to boot. The drive should appear in My Computer and be accessible.
If this works you can copy data from that old HDD to another device. Maybe some newer HDD that fits into the external enclosure would do. Then you can later copy the data again from that intermediate device to your newest computer.
I have assumed here that your old Gateway machine does not have any way to access a network. If it does, there may be an easier way. Put the old HDD into the Gateway where it works and connect that machine to the network. Set up access privileges so that the old Gateway and its drive is available to other machines and simply copy data cross the network.