It may be possible through some third party devices, but there's no legitimate way to do so.
I know that the "Mega Memory" cartridge for the Game Boy could store the saves, but I have no idea if it could upload it to a PC. I do know that the Action Replay for the GBA could do that for GBA games, but not for original Game Boy games.
Where logic won't work - I will PAL Gamers unite - http://www.gamefaqs.com/boards/277-pal-gaming
If you have a Game Shark with a save state feature, this is doable. I saved a very glitchy pokemon yellow game in progress that way once. You need to save state the game in play, then hook your game shark up to your pc with the included parallel cable(an old school printer cable). Some of them come wiht 25 pin serial cables instead, and some come with nothing at all. Maybe they did a USA version near the end of the run, but I've never seen one.
The effect is just like a save state in an emulator. You save in the middle of playing, and it does not utilize the in-game save feature, unlike the mega memory card. After saving to your computer, anyone with a gameshark and the same game and the cable can download and load up your save right where you left off. You can with just the shark load the save onto another game pak. You can even load a save from one game onto an entirely seperate game, sometimes. Try loading a blue save state onto a red cart, for instance.
I Wish more people wanted to play Video Games with me. Jeez, just mail me the game and I'll beat it for you. - BubbRubb
There's an item called a "GB Transferer II" (just Google it, and you'll find it) that lets you import your saves to the computer. IMO, that is better than using a Gameshark because you are not forced to stay with Windows 98 (you can use XP, Vista, or 7). I bought it recently, and it has perfectly backed up the saves of my six Pokemon GB games.
Keep note though, because you'll need a computer with a parallel port to use it.