Are you asking about storage of a dilution, or storage of the reagent-grade (>98%) compound? If you're just storing the stock from a manufacturer (or an aliquot of that), just throw it in a 4°. At a previous lab, our beta-mercap was kept on the shelf at RT and we never had an issue with it, at least during the 4 years I worked there. But it is recommended to store at 4°C.

For a dilution, definitely store it at 4°C. I've read about people freezing it at -20°C, which is fine for long-term storage. If you only need to be using it over the course of 8 days, it should be fine at 4°, and actually I'd be hesitant to store a working aliquot at -20° since you'll be freeze/thawing it multiple times. Note that the more dilute a sample of BME is, the more quickly it degrades. Keep it at the highest practical concentration you can.

If it stops smelling like hot garbage, don't use it.