What are LEGO rubber bands made of?
They are made of SBS rubber.
According to user &quot;B. Lister&quot; on eurobricks.com, these bands are silicon rubber, used in medical and dental work for long lasting connections and tension.
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However, a LEGO representative at a toy safety site indicated otherwise:

The colors and materials in ABS plastic do not rub off or leach and, in fact, meet the same material standards for safety that are required of plastic eating utensils. Tires and elastic materials found in some LEGO sets are manufactured from Styrene-Butadiene Styrene (SBS), a material commonly found in vehicle mud guards and shoe soles.

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SBS has some very similar traits to silicon rubber, including sterilization and long lasting properties. I noticed in the other answer that bike tire cuttings worked almost identically. They should! Those inner tubes also consist of a Butyl base.