You may be wondering why I ask such a question.

Well, I bought a Samsung 2.5" 1TB HDD a few months ago. It was running fine but a few weeks ago I heard a dynamic beep (non monotone) and this was proceeded by a very audible click. My first guess at the cause was the HDD head becoming stuck which will cause both the beep and the audible single click as it quickly parked the drive head but this happened while I was using the HTPC normally, it was as if nothing had happened and everything was running fine (me browsing the internet). Would such a HDD failure crash Linux (ubuntu in this case) instantaneously if it were in use at the time or would it not matter?

Ever since, whenever I shutdown my HTPC it makes this audible click which it didn't make before, if it is indeed the HDD which makes this noise I'm guessing its autopark function is now broken and I should be looking for a replacement.

You would start seeing error messages very fast I think.

Haha, if you can page file somewhere else. And don't have open files that are actively doing i/o mail, avi, mpeg from the disk. No.

I think it should work if you set it up not to use swap space and got enough memory. Though I don't know how would you launch programs as there is no hdd to read from. manipulating root would also be inpossible I guess.
Anyway whatever you load in the memory should stay there as long as it's needed(programs running) or system restart/shutdown.
Try to opt for a lightweight setup if you can(xfce,lxde,or no windowing system).
HDD is not needed for a computer to work, though it's helpful, I remember when we used to boot from floppy or network (DOS).

Please note that I'm a Linux noob.

Live CDs / DVDs exist with a reason. They are specially prepared to load everything into RAM.

Puppy Linux work well with no HDD at all

see also Damn Small Linux and LFS.

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

Thanks don_falcone, on a live CD the OS is loaded into ram but I'm not sure what happens when you remove it either. They are not persistant for a reason and any additional software you run is still loaded off the disc. So pulling out a LiveCD mid-use is one thing especially when the applications know that there is no write access and no persistance when they load into ram from a liveCD, suddenly yanking out a HDD however... I'm even less sure at what happens.

I haven't tried pulling out a LiveCD while using it, I'll try that later (I use LiveUSBs since they offer persistence) but I suspected that removing a LiveCD when the OS is running wilI still throw up an error and more than likely prevent the use of any further applications (they are not loaded into ram and reside on the disc only) since you essentially removed the entire filesystem. just as I suspect yanking out a HDD would do even worse. I just didn't know Linux would handle such a failure so gracefully and without warning and wanted to determine whether my HDD was failing or if it was something else. It seems odd to me that Firefox didn't complain about its directories suddenly disappearing, Linux suddenly losing its filesystem, etc.

I'll check the S.M.A.R.T log of the HDD as suggested, I remember my old drive in this cramped case having a few warnings that the temp threshold was exceeded, I'll see how this one is holding up.
I have all my data on a NAS device so I'm not too worried about its failure.

You could use the NAS for your system minus the swap space, just need a usb stick for a bootloader. I think it's possible, though never done it myself. You could even use a virtual file system like WUBI uses. Maybe it's easier just to get a new HD. :P

If you take out a Live CD the OS will still work, up to the point it needs to acces a file on the disc. When that happens the OS would ask you kindly to put back the damn disc. I have done it many times with winxp install cds :P(I got a few and sometimes had to switch them around on the fly)