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Space under the stairs is easy to ignore because it's so well hidden. If the stairs aren't carpeted, carefully pry up a tread and use a flashlight to look underneath. If prying up a tread isn't an option, cut a small inspection hole in the drywall – it's much easier to patch a small drywall hole than replace a large piece.
If the space checks out, knock a larger inspection hole between two wall studs and take a closer look. Most stairs are well-supported at the top and bottom of the stringers. Make sure the stairs have this kind of stability before proceeding. Check with a structural engineer to determine if the wall is load-bearing – never remove studs from a load-bearing wall.
Locate the studs in the wall and determine the position for the closet door. Use a reciprocating saw or a drywall saw to cut the drywall – but not the studs – to accommodate the closet door.
Safety Alert: Reciprocating saws are powerful and can have a strong "kick." If you're not confident you can control the tool safely, get assistance or use a different tool.
Score a level line from the closet hole to the bottom of the stair stringer and cut the drywall along this line. Also cut away the drywall along the bottom of the stair stringer.