I have no idea what my wall is made of, but I can't get into it with a
standard drill and bit or with a "self-drilling anchor". All I
managed to do is gouge out a hole in the wall. Any idea what it may
be made of, or how to get into it?

Another wall has a similar problem: I can't hammer a nail into it
deeper than a coupla mm. Presumably made of the same stuff, but this
time I didn't try much harder than that to get into the wall.

Thanks,

Michael Hamm
not much of a DIYer (obviously)

http://www.math.wustl.edu/~msh210/