Gastown is Vancouver's oldest neighbourhood and features cobblestone streets, galleries, cafes and boutiques. There are a few tourist trap-like places but the architecture is interesting to see. The Gastown Clock is a working steam clock, one of only a few in the world.
Important heads up - I would recommend avoiding walking east of Cambie Street while exploring this area. The The next 5-6 blocks (Abbott, Carrall, Columbia, Main along East Hastings and W Cordova St) are a particularly bad section of the Downtown Eastside, an area of the city with high homelessness, drug use and poverty. I've never felt truly unsafe here but I have certainly felt uncomfortable.