I own a barge on the Seine in which I live. I would like to sell it. Can I claim exemption from capital gains tax on the sale?

Yes, under certain conditions. It is possible to benefit from an exemption from capital gains tax when the property sold constitutes the seller's principal residence on the day of sale (art. 150 U CGI). For tax purposes, this residence can only be a residential property. Legally, a barge is a movable asset (art. 531 C. civ.).
However, by way of derogation, it is accepted that the sale of a houseboat for residential use can benefit from this exemption if all the following conditions are met: 
   - the barge must not be intended for navigation.
   - it must be subject to property tax on built-up properties. 
   - it must be used as the owner's main residence on the day of sale, at a fixed location (BOI-RFPI-PVI-10-40-10 §320).

Note: this solution also applies to boats (BOI-RFPI-PVI-10-40-10 §320).