As Jimbo and Troy have pointed out, there are any number of ways and software available to back up data.
If I follow you, I think this would be easiest and the cheapest…as in no cost to you.
I have no idea of the OS, ie., operating system(s) you’re using…windows XP??? or motherboards etc…are these proprietary boards, like dell, hp etc.
In any case, install the larger hdd from the “obsolete box” into the new rig.
Be certain to place this as MASTER on the IDE cable (flat ribbon cable) and be certain the jumper is correctly set to master (you may have cable select anyway…I have no idea about your set up)
Now install the small hdd into the new rig and connect this as SLAVE…again, be certain the jumpers and ribbon cable are correct.
Format the slave drive and then simply drag anything you want off the larger hdd to the slave hdd.
Now you can format the larger hdd and install a new OS.
You can keep the smaller hdd right there and use it as a slave…extra space for you…or you can transfer whatever data you want back to the larger hdd and do whatever with the smaller hdd.
This way, you keep all your options open and you can take your time figuring out what you want to do.
Were you planning on keeping the old system running?