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???
He might have an on board toilet
It was a cold day near the southern base in Blood Gulch. Cortana and I were relaxing over a game of Go Fish, WHEN OUT OF NO WHERE 700 BANSHEES CAME FLYING IN!!! I GRABED MY SPARTAN LASER AND LET HELL REIGN DOWN UPON THEM. I HOPPED IN THE NEAREST WARTHOG AND TOLD CORTANA TO GET IN.
Magic.
He holds it for an hour while someone takes his armor off.
They have eliminated the need for it. I'm guessing with all of the Spartans augmentations and the kind of food they eat(which they don't do often) all of it is stored and there are no wastes to be produced.