I came up with this rather amusing and bizarre stalemate where Black stalemates itself by self-smothering.
[FEN "8/8/8/8/8/3p4/pp6/krb1K3 b - - 0 1"] 1...d2+?? 2. Kd1
The peculiar feature of this stalemate is that Black is unable to move not because moving a piece would put its king into check, but moving a piece would require it to capture its own piece, which is disallowed. This is an example of a single self-smothered stalemate.
The question is, is it possible to have a double self-smothered mate, such that both sides do not have any moves at all because the moves would require capturing of the side's own pieces?
One possibility suggested by the user supercat is the following -
[FEN "KN1b/RPpPp/P1P/4P/8/3p1p1p/3PpPpr/4B1nk w - - 0 1"] 1. e6
And both sides are stalemated. However, this position cannot occur in practice, because you cannot have a Black bishop trapped on d8 with the c7 or e7 pawns still on their original squares.