First of all, you should know that there’s no such thing as “unlimited disk space”. The more you pay, the more space you get.
Second, you said you have an old computer, therefore I doubt if it supports eSATA. Speaking of internal or external hard drives, you’d better get an external one, even if it’s a bit more expensive than internal drives. The reason is that since you already have an old hard drive installed on your motherboard’s BUS, your motherboard will probably end up having issues with the old and new one installed on the same BUS, it may not even recognize the old one. But if you get an external drive, you’ll connect it through USB (or SATA if your motherboard supports) which is also a lot faster and your system won’t have any problems detecting those different hard drives. Hope it helps.