I assume you by "stable" you mean that the DNA is not being degraded during storage. Stool samples will wary between 'donors' and hence so also the concentration of DNA damaging factors. I would suggest that you verify for some of your samples that 4 degree Celsius storage does not result in DNA degradation by comparing a series of gel-runs with extracted and purified DNA obtained from a stool sample as early as possible.
I agreed with previous replies and there shouldn't be any problem in storing at 4 C. I have sample nicely stabled for 2 years so far personally. However for storing for many years better to put them at -20/-80