Closer than was known by the US public at the time, but not so close that they would have had a realistic chance of succeeding.

The USSR had started making plans of a manned lunar programme shortly after president Kennedy announced the US one. As the USSR had no superordinate space agency like NASA, different mission profiles and technical solutions were proposed and worked on by competing design bureaus, all vying for political support, which was the only way to receive funding. In the end the mission profile chosen was quite similar to the one chosen by NASA: A single launch with a large booster rocket, inject into lunar orbit with a crew module and a seperate lunar lander, and afterwards leave the lunar lander behind at the moon. However, rivalry between design bureaus never stopped and competing projects were in development until 1965. This fragmented funding, which together with the generally lower economic power of the USSR as compared to the USA caused the moon programme to be seriously underfunded. The only chance Soviet designers had was to use improvisation and tried technology, cut corners as much as possible and hope for the best.

The development of the main components was given to different groups who achieved varying levels of success. The crew module was the only one developed successfully — after the end of the lunar programme it found new use as a general-purpose crewed spacecraft. We know it as the famous Soyuz capsule still in use today.

The lunar lander (called the Lunniy Korabl — "lunar ship") was very similar to the one NASA eventually used, but smaller: It was intended for only one cosmonaut, which caused additional problems, such as the need for him to be able to put on his space suit without help. Development of the LK went on into 1971, and in the end it was considered ready for use in a moon mission (which never happened). Space suit design also was successful, but found no use in the end.

The part of the mission which ultimately doomed it was the biggest part of it — the development of the booster rocket, named the N1. It fell to Sergey Korolev's OKB-1 (design bureau 1). The central problem was that developers felt they neither had the time nor the funds to develop a huge rocket engine to power the N1's first stage. A moon rocket for the mission profile chosen by both Soviets and US (Lunar Orbit Rendezvous) was necessarily an order of magnitude larger than all the rockets in production until then. NASA had the good fortune to have the huge Rocketdyne F-1 engine available, which had already been developed in the late 50s. The Soviets had no such thing, so Korolev opted to use a clustering of smaller NK-15 engines — 30 of them. Those were an untested new development as well, by a designer who had no previous experience with rocket engines (his design bureau had worked on air-breathing jet engines until then). The result was a design that one commentator called "a plumber's nightmare". Also constraints in time and money made it impossible to test most of the engines prior to being assembled into the rocket. In this situation it was a given that one or several engines would fail at every launch. To remedy that a complex computer system was installed that would detect such failures and switch off the affected engine as well as the one directly opposite to it, to retain balance. It turned out that it was this computer system which caused most of the test launch failures. In the end the N1 was a technological dead-end and could probably never have worked properly.

In 1966 Korolev died unexpectedly and basically could not be replaced, especially his political astuteness in providing funding. Design of the N-1 went ahead for a few more years; four test launches were performed between 1969 and 1972, each one resulting in failure and sometimes catastrophic explosions.

After NASA had landed on the moon in 1969 there was no point in continuing any more. The "first" had already been achieved, the cost of the program was high and it was not even close to success yet and a moon rocket has no military use (e.g. as ICBM or spy satellite launcher). In the end the program fizzled out (in 1974) and the USSR cut their losses and set their sights on other space projects which had not been achieved yet (and which provided a convenient cover for military activities): space stations. It possibly helped that the lunar programme had not been publicly known, so there was no loss of face involved: e.g. I have a book on the history of space exploration from the 80s which only contains a single sentence about the whole Soviet lunar programme; there was not more information publicly available until the end of the Soviet Union.

The history of the Soviet space programme is a long and fascinating story, and I could only provide rough outlines of a part of it here. I definitely recommend looking into it in more detail!

EDIT: This website has a great collection of quotes by the leading Soviet space designers of their own assessments as to why the project failed.