Yes, it is possible to turn gore off.
1. Go to the Library tab on Steam
2. Right click Team Fortress 2 from the game list
3. Click 'Properties' from the dropdown menu
4. Under the 'General' tab, click "Set Launch Options"
5. Enter "-sillygibs" (no quotes) into the box, then press OK
Now your gibs will be replaced with gears and springs and little toys with no blood.
note: Blood will still show up on models if hit by bullets, but explosive kills and damage will not spew blood everywhere like they do with real gibs.