Can a person claim exempt on their tax forms when being hired if they will be below a certain income level?

In a way, your thinking it through too far -

You claim the number of exemptions that would be realistic. The tax rate for your expect income, (applying those number of exemptions), is what payroll will withhold. If that is 0 because of you pay level, then nothing is withheld.

Your not really exempt from tax (and I'm not sure there is an election called "exempt" anyway), you just don't have any tax liability. Of course, even at a low income level you wouldn't be "exempt" from paying taxes like...FICA, Workers Comp., Disability, Unemployment insurance, etc. that ave not minimum base.