This is actually sort of a difficult question because the origins of the words used to describe Grendel's home are unclear in the old texts. Translated, the text says something like this...Grendel is:
"in moorland living," (from chapter one)
A moorland is a land full of moors, which is sort of like a boggy, scraggly, inhospitable place. There is reference to "Grendel's Mere", "Grendel's Pit" and "Grendel's Peck" in the Anglo-Saxon Chronicle. These would lead the reader to believe that Grendel is living in some kind of underground pit, watery cave, or nasty hole within a swamp, though it never really says. It's a safe bet, though...Grendel's mother does indeed live in some kind of underwater cave in the swamp, so this probably would support the idea that Grendel lived in one too.