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My chemistry class is working with fuel cells. And I am part of a
project that uses a Stirling Engine to clean the fuel that goes into
the fuel cell. So...
How cold does a Stirling Engine get?

Heat (energy) is applied to the hot end of a Stirling engine which produces some mechanical power output and rejects some heat energy at the low temperature end. I believe you are interested in a Stirling cooler which is driven by mechanical energy, draws heat from the 'refrigerator' and rejects the equivalent heat of the mechanical energy input plus that removed from the 'refrigerator' to a higer temperature heat sink.
Help me out with more information.
I assume that you want to process "a chemical fuel of some kind" at a low temperature to purify/prepare it for use in the fuel cell.
What temperature is required? What is the chemical/fuel? How much needs to be purified/processed?
Is this a real live practical project or theoretical?
Do you expect to build or buy a Stirling cooler?
Do you have any idea as to what is involved in building a Stirling cooler or the cost of buying a commercial machine?
Specifics will help to provide useful answers.