What are the best Denver neighborhoods that are close to downtown but do not feel like downtown?

Congress Park is a great neighborhood with a lot of wonderful bungalows and townhouses. It's a stones throw from Cherry Creek restaurants and shopping but also has neighborhood restaurants and coffee shops within walking distance.

Stapleton is a unique neighborhood close to down town with a community feel.

Of course it depends upon what you mean by both "close to" and "do not feel like downtown." Denver is a large metro area but within just a few miles of the central business district there are many neighborhoods that have a small town feel (with a block or two of their own neighborhood markets, restaurants, service business, etc.) and are easily accessible by bicycle and public transit routes. They each have their own personality and honestly feel like a small town (sometimes even a village) within the boundaries of their neighborhoods. My favorites with everything you could want in a small town feel are Park Hill, City Park, Congress Park, and Cherry Creek. As their names imply they are all either located adjacent to a major park or recreational area while still being just a few minutes from the hub and hustle of downtown.