Is it possible to put video files on a PS3's hard drive for viewing and storage? Specifically video files from a computer, not videos downloaded from PSN.

Yes, it very much is possible. Just put your videos on a flash drive and plug it into your PS3 and copy it.

Yes you just have to put it on a USB stick (or PSP) and then go to the video tab on the XMB. It should list a USB drive (or PSP) which you press "Triangle" on and choose "view all" or something similer (with an all) which will let you navigate through your USB stick to where your video files are stores. When you find your video files you press "Triangle" and then "Copy".

I imagined that it worked like the Xbox and was read only for media. 
 
Can you copy them back from the ps3 after that? No issues storing bunch of video files on your ps3 hard drive?

Ok, that clears it up. Thanks everyone!

You can just make your PC a media server and stream files from it to your PS3 if you got a network setup.

For some reason flash drives never worked for me so I just used a program called PS Media Tunnel to get music, pictures, and videos on my PS3.

The problem with using a USB drive is that it must be formatted in FAT32, and therefore file sizes cannot be larger than 4 GB which can easily happen when working with HD video. The two solutions I've found for this are splitting the files or I prefer transferring them over the network. I don't know how you would get files larger than 4 GB off of the PS3, though.