The Pickett House at 910 Bancroft Street is the oldest house in Bellingham, built in 1856 for George Pickett — then a US Army captain commanding Fort Bellingham. It's a simple white clapboard house that has survived remarkably intact, and the reason it draws visitors is entirely down to what Pickett did five years after leaving it: he led the famous Confederate charge at Gettysburg that bears his name.
The house was preserved specifically because of that later notoriety, which creates an interesting tension — it's a Civil War memorial in a place that had nothing to do with the Civil War. The structure itself is modest, but the combination of age (it predates Washington statehood by 33 years) and backstory makes it worth the short walk up from downtown. It's maintained by a local historical society and occasionally open for tours; call ahead to confirm access.