Op-amp output wanders over the course of a few seconds
Welcome to the world of low frequency noise!

If you really want to achieve the 0.8uV pk-pk noise at a gain of 1000 listed in the datasheet over a long term, you'll need to temperature control the part at 25C (like they did, as listed in the datasheet). It looks like even after subtracting out the long term temperature noise your at about 0.005mV pk-pk/991 = 5uV pk-pk which is still 5-6 times more than the factor listed in the datasheet. 

You'll also need to not have any further opamps in the signal chain, as they will increase the noise. Your ADC will also add noise depending on what it is. 



If 1/f noise is killing your application, build an instrumenation amplifier from chopper amps, or find a chopper instrumentation amplifier.