Popovers on the Square for informal coffee and conversation.  Or over breakfast.

Lunch: Rudi's, just off of Market Squareat 20 High Street is a great setting for amid-scale lunch.  Quiet and efficient.

While open to members only for traditional dining, if you are looking for the ultimate Power Lunch or corporate dinner and are in need of space for media and/or conference table situation, you can request a private room rental at The One Hundred Club.  Just one block from Market Square, The One Hundred Club offers fine dining and the most incredible views in all of Portsmouth.