Landfills are basically now the worlds largest plastic bags. Basically a least a foot of clay at the bottom (which liquids flow slowing through), then sheets of HDPE plastic welded together to make a (hopefully) watertight surface. Usually they have pipes to collect whatever leaches through the trash. In NY being not trusting sorts... they double line their landfills with 2 layers of liner.

Then when its filled you put a layer of plastic and soil on the top to stop water from getting in.

Its not perfect, but it seems to be ok. Nothing rots in there since there is hardly any moisture and the off gassing is who knows what (sometimes they burn it, so it has some methane).