I have strength of 4 and some weapons (like the hunting rifle) are str min of 5. When I shoot, I still hit with good accuracy, and tend to do decent damage, but the info window always says "you don't have enough strength to use this weapon properly".

Exactly what kind of penalty is being applied for using a weapon under it's minimum strength rating?

It's a 20 percent penalty to accuracy for every strength point missing if I remember correctly, if your weapon skill is high enough you can still hit with the capped 95 percent accuracy in spite of the penalty.