It might be possible…just maybe. While not specifically excluded by the National Electrical Code (NEC), there are two code standards that a panel must comply with to be located under a stair:
1) The under-stair space is usally a closet, and a panel cannot be located in a closet near ignitable material [NEC 240.24(F)]. So, if clothes or other ignitable materials are stored in the closet, it’s a no go.
2) Also there must be a clear space in front of the panel to safely service it, which is defined by code as 2’-6” wide, 3’-0” deep, and 6’-6” high [NEC 110.26 (A)(1,2,3)]. There is rarely that much clear floor area in an under-stair closet for a panel, or that much headroom. 

    And an appliance in the closet, such as a water heater, cannot infringe on the required area. For more on this requirement and its history, go to When did the requirement for clearance in front of an electrical panel become code?

    Also, see our articles Can an electrical panel be located at a stair landing? and Can an electric panel be mounted over stairs?

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Field Guide for Home Inspectors, a quick reference for finding the age of 154 brands of HVAC systems, water heaters, and electrical panels, plus 210 code standards for site-built and manufactured homes, and the life expectancy rating of 195 home components. Available at amazon.com for $19.95.
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Here’s links to a collection of some of our other blog posts about ELECTRIC PANELS:

• Can an electric panel be located in a closet? 

• Can an electric panel be located in a bathroom?

• What causes copper wires to turn green or black in an electric panel?  

• What is the maximum number of circuit breakers allowed in an electric panel?

• When should a corroded or damaged electric panel cabinet or disconnect box be replaced? 

• What is a tandem circuit breaker? 

• When did arc fault circuit interrupter (AFCI) breakers first become required? 

• Can you add circuit breakers by different manufacturers to an electric panel if they fit?

• My circuit breaker won't reset. What's wrong?  

• What is a split bus electric panel?

• How do I identify a combination AFCI (CAFCI) circuit breaker? 

• What does a circuit breaker with a yellow or white test button indicate? 

• What is the maximum gap allowed between the front of a recessed electric panel box and the wall surface surrounding it? 

• What are the requirements for NM-cables entering an electric panel box?

• Why is a fuse box/panel an insurance problem for homebuyers? 

• Why is bundled wiring in an electric panel a defect?

• What is the difference between GFCI and AFCI circuit breakers? 

• What does it mean when a wire is "overstripped" at a circuit breaker?

• Why is an old fuse panel dangerous?  

• Who is the manufacturer of those "bad" electric panels?

• Why is the circuit breaker stuck in the middle? 

• What is a double tap at a circuit breaker?

• What is the right electric wire size for a circuit breaker in an electric panel?

• What is the life expectancy of a circuit breaker? 

• What is the maximum height you can mount an electric panel above the floor? 

• What is the main bonding jumper and where do it find it in an electric panel? 

   Visit our ELECTRIC PANELS page for other related blog posts on this subject, or go to the INDEX for a complete listing of all our articles.