My guess is Windows would be unable to format it even if you wanted it to.
What often happens with external hard drives is that the USB converter board inside the enclosure fails.
This causes a communication error between the drive & your system, which means Windows can't read it properly, nor can it write to it (which is why attempting to format it will fail too).
I've never had to do this myself (I have four backup drives "mirrored" with the same data, so I just sling it and buy another when one fails) but I do know for sure that removing the drive and putting it into a third-party enclosure has done the trick for many people (I help on a lot of forums).
However, a failed converter board is not the only possible cause of your problem. It's a hard drive, and all hard drives fail eventually, especially those which get carried around regularly. So I recommend you try testing it with "SeaTools for Windows". I say "try" because SeaTools may even have problems accessing it.
Get "SeaTools for Windows" here: http://www.seagate.com/support/downloads/seatools/
Finally, a word of advice. Saving your data on only one hard drive (and especially an external one because they are less reliable than internal ones) is very unwise. All hard drives have a limited and unpredictable lifespan and all will fail eventually.
If all your data is saved on only one drive, that means you don't have any backups!!!
You should at all times have identical copies of your data on at least two drives, so that one acts as a backup. When a hard drive fails completely, not even the best tech support in the world can work miracles to save your data.