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Yes, you could make this into what you describe. The need for a pump depends on the topography of the lot
If the area can't be drained naturally or with a French drain you'll need a pump
Cost depends on the site conditions
Is this something a local handyman service would provide, or a major chain like Lowe's? (There is an existing french drain and sump pump in basement)
Is the existing opening only big enough for a single door?
This is a job for a general contractor not a handyman or a big box store
Existing opening at bottom for single door right now so that would have to be expanded but stairwell and excavated area is wide enough for double door
does the lot slope away from the house to a point lower than the basement floor?
where is the house? what State?
Ok, I'd say you're looking at around 10K depending on what type of stairs you plan to install and assuming the excavation walls are adequate the way they are now. This would mean that you would have to drain the bottom of this area into the basement sump pit. It sounds like conditions would permit a French drain which is a better option but hat could add several thousand to the cost of the project.
Thanks Rick appreciate your help. Signing off now.