More detail to check on drive size limits.
The original LBA system for disk access used a 28-bit address which allowed up to 137 GB (Microsoft calls this 128 GB) to be used in a drive. The newer system, called "48-bit LBA Support", came into use around 2000, so your machine is in the borderline area. Older systems typically said they had "LBA Support" but did not specify the "48-bit" part.
48-bit LBA support is needed in three places - the hardware BIOS that runs the hard drive controller, the hard drive itself (in its own on-board controller) and the Operating System. Any hard drive made over 137 GB obviously will have it, so you need to check the other two.
On the OS side, Windows XP original did NOT have 48-bit LBA Support, but it was added in SP1, which you say you have in the Install disk you plan to use, so that's OK.
So, does you machine already have 48-bit LBA Support in its BIOS? I could not tell from scanning their website, but there is a reasonable chance it does. If it does not, it also is possible that Gateway may have released for it an updated BIOS you can download and burn into your BIOS chip to add this functionality.
If it turns out you cannot get your hardware to support disks over 137 GB, there is still a route for you. I've done this in your situation. Seagate has a feature in some of its drives that allows you to use their free software utility, Seagate Seatools for DOS, to force the drive to limit itself to that so no error can occur. So, for example, I bought a Seagate IDE drive of 160 GB capacity, then used the tool to tell it to behave as if it really were only a 137 GB drive. No disk controller or OS will ever know otherwise. The way it's done is that, using one part of Seatools for DOS, you must set the maximum number of disk Allocation Units or Sectors allowed. The number you enter is just 2^28, or 268,435,456. At 512 bytes per sector, this gives the limit of 137,438,953,472 bytes. With this hardware adjustment made on the drive, you are guaranteed safe using it in a system with only 28-bit LBA support.
Oh, by the way, sometimes there is an unrecognized space for another hard drive in a case. The slots used for 3½" floppy drive mounting are the same size as the ones for 3½" hard drives; the difference is simply that the floppy drive bays are located up near the top and have a removable cover so you have direct front access. Although you have one floppy drive installed, is there an empty floppy drive bay under it? If so you can leave its front cover in place and mount a new hard drive in the space.