The default answer is you'll run into problems immediately if your HDD dies. The normal behavior will be that your whole system will hang or crash.

There are some means to work around a non-working HDD. The best known way to boot from a live medium like a CD or USB key has already been mentioned. Another way would be to load the whole system into the RAM during the boot process. This requires the HDD to work properly during boot but it can fail anytime later. Distributions based on Linux Live Scripts [1] support this feature (Slax, Porteus, PureOS), Ubuntu does not.

bottom line:
If your HDD starts to show signs of failure you should replace it as soon as possible regardless of the OS you use.
smartctl from the smartmontools package [2] might provide some detailed info about the HDD's health status.

edit:
By default Linux does not use page files but swap partitions.

[1] http://www.linux-live.org/
[2] http://packages.ubuntu.com/oneiric/smartmontools