The local Area Research Center may hold microfilm of deeds with their grantor-grantee indexes. A very useful record is the city assessor's card , kept by some assessors' offices. Eau Claire's date only from 1953, but contain earlier information gathered from neighbors or other sources. They may contain floor plans, diagrams of additions, date of construction, etc. You may be able to approximate or even pinpoint the date the house was constructed from this card.
Building permits are to be found at the city building inspector's office.
However, they may not date very far back, perhaps only to the 1950s or '60s. They may be useful for tracking modifications to a house, but unless your house is fairly recent, not for establishing date of construction. The Eau Claire office has put the permits, dating back to the 1950s, on cards, filed by address. The cards provide the date, a written description of what was done, and the contractor. In Eau Claire it is in the Special Collections Department of McIntyre Library at UW-Eau Claire.