From Dubai Museum, one can just stroll to the Bastakia Quarter in Bur Dubai. This is a historical place, and maintained by the Dubai Municipality. There is no entrance fee. There are several shops selling artisan art, trinkets, and souvenirs...and a coffee shop that offers camel burgers (around AED45-bit expensive, but you're eating camel - so I guess, you have to pay for the experience), and art galleries. I suggest you allow a minimum of two hours exploring the Bastakiya quarter because they have many permanent exhibitions - such as an extensive coin collection. It is also a vibrant place during art festival months because of the free art workshops, performances and art exhibitions.