As a space nerd I can attempt to answer this:
The [N1 rocket](https://en.wikipedia.org/wiki/N1_(rocket)) was in development when the moon landing occured. It had made two test flights by the time of apollo 11, both ending in failure. It later attempted two additional test flights (also failed) and was eventually cancelled.
The failure of the N1 means they were not particularly close. Presumably with continued funding they could have landed on the moon as well, but they were several years behind. The N1 rocket had less performance than the Saturn V so a soviet moon mission would have been a smaller afair due to mass constraints.
The NK-33 engines developed for the N1 program are very high performance and are actually being used today to power the american [Antares](https://en.wikipedia.org/wiki/Antares_(rocket)) rocket which is sending cargo to the ISS. The engines were refurbished/rebranded in the US but the original "metal" dates from the 60s/70s.
This thread should be relevant to your question. http://www.reddit.com/r/AskHistorians/comments/2bcerc/when_apollo_11_landed_on_the_moon_how_far_away/
hi! you may be interested in this section of the FAQ (link on sidebar) - it includes one post directly related to your question, while the rest will help provide context on the Soviet space program