beet stock shelf life
Beet stock (the cooking liquor left over after boiling beetroot) contains plenty of nutrients, and hence it's a good breeding ground for mould and bacteria.

I would treat it like cooked vegetables, and throw it on the compost heap (or down the drain) if not consumed after a week in the fridge.

Of course it could be preserved for longer if steps were taken to combat mould/bacteria:


reduced to a syrup, perhaps with added sugar
with lots of added salt, as in a pickling brine
sterilised in an airtight container