Is it possible to play with the PS3 without the Hard Drive? I mean, like having and usb drive with the OS and some saves from the games? My Hard Drive is dying, I think, and I want to know if this is possible.
Games tend to take up a lot of space nowadays. Uncompressed and high quality video, audio, and textures eat up gigabytes of space; throw DLC packages in there, and you're looking at upwards of 80+GB for one game (I'm looking at you, Call of Duty.)
A flash drive can likely be formatted to hold the PS3's operating system and any games, but the limited speed of USB 2.0, and the fact that flash drives tend to be small, means you won't be fitting much at all, and if you do, it will be as slow as molasses.
The recommendation here is to buy a new internal hard drive. There are plenty of guides around (TechRadar has a good looking one), and the process should be relatively easy. Just make sure you get the proper physical-dimension (2.5 inch Sata) drive for the internal bay, as even the slightest change in size can render a new drive unusable.