I agree, some piece of hardware, perhaps including the Windows CD is faulty. RAM, HDD would be the first things I'd check.
Testing the RAM: www.memtest.org
Testing the HDD: all HDD manufacturers have free downloads of diagnostic software, so you need to find out what HDD you have.
If you want all-in-1: get the "Ultimate Boot CD".
Later, when you successfully installed your Windows, go to Windows Update website and install all available security updates including Service Pack 3. Windows XP without SP3 is out of support now.
Another hint: create restricted user accounts for the kids and assign a secure password for the account that has admin permissions.