Yes, replace it immediately after you finished backup up as many files you can into a CD, DVD and/ore flash drive. You also have to get ready to install a new one by reinstalling all the system drivers, Operating System and any app that's related to the drivers like Catalyst for some nvidia cards.
A harddrive is very very sensible equipment. Believe it or not, it is the most fragile part of a computer than a monitor. The needle-laser that reads/writes the data into it's clusters is the equivalent of a Boeing 747 flying at mach 10 1 inch off the ground. Just takes 1 tap to make a chip which bounces around inside while the drive is rotating at over 7200 speeds and more.
Imagine the damage and cumulative wear from 1 tiny piece of medal hitting that disc more times a human can count from 1 to 2 in miliseconds 100 times.
However, even if it's not hardware issue bad sectors can also be a driver issue related to the HDD like IDE/SATA etc drivers missing or the power cable or the data cable not fully plugged in. Bad registries can also create bad sectors.
Before replacing the HDD try to defrag the computer, clean it up and clean the registry. Best free cleaner out there is CCleaner. You can google for it.
If that does not help, the physical and software aspects, then you have no other options but to rule in a damage HDD.