There are conflicting accounts of this, but here's what Robin Frost says in his Adelphi Series volume ("Nuclear Terrorism after 9/11", chapter 2):

The nuclear project began in earnest in 1993, when the cult bought a number of mining leases on an Australian sheep ranch with a known uranium deposit. It also explored the feasibility of buying a ship to move uranium ore overseas. Earthmoving and rock-crushing equipment was brought to the ranch, and a small laboratory was set up. Although its activities during its short tenure on the ranch remain mysterious, Aum is known to have tested sarin gas on some of the sheep there. Also in 1993, the group tried to buy a sophisticated laser measuring device (an interferometer), designated as ‘dual-use’ by the US government because it can measure, with great precision, the velocity of high explosives or the surface of a uranium or plutonium sphere for use in a nuclear weapon. Aum also attempted to buy a vibration isolation table that could be used for the same purpose. However, the manufacturer became suspicious and alerted the US export authorities, and neither sale was completed. The cult also tried to buy a $450,000 laser welder from a Californian company, indicating that the machine was to be used in a sealed room and operated via a glove box, which would only have been necessary in the presence of hazardous materials, including radioactive substances. When ‘a member of the cult’s Science and Technology Agency’ was arrested in March 1995, he was carrying a document about laser enrichment.

Assuming that is correct, they were not close at all. As Frost goes on to explain, laser enrichment is not easy to make work (even now, much less 20-30 years ago, when it was still very much emerging technology), and raw uranium doesn't do much unless you also have a lot of other facilities: you need to mine it, mill it, and then either convert it to UF for an enrichment facility, or turn it into fuel for a reactor; if you go the reactor route, you need hot cells to process the plutonium out of the (very radioactive) spent fuel, need to do something about the other waste generated. If you end up with a significant (e.g. 5-10 kg) amount of plutonium after all of that, yeah, you can turn that into a weapon, but implosion is a pretty tough engineering challenge, and apparently Aum did not (by the time it was discovered) have significant investigations into those (e.g. the use of shaped charges). If you go a uranium route, a simple bomb is much easier, but it still requires you to enrich the uranium, which is very hard to do.

What is interesting about them is that they appeared to have been trying to create a full nuclear fuel cycle, which is not the "standard" assumption about nuclear terrorism, for the obvious reason that it is difficult, expensive, and creates many detection opportunities. (It's not impossible to imagine private companies with the capacity to do this — but they would have to be huge. Bechtel could probably pull it off, but not without being detected.) Whereas stealing fissile material, while it still presents many difficulties, is at least something more readily contemplatable by a non-state entity (if Aum had 50-60 kg of HEU, they certainly had the resources to turn it into a weapon, but so could any decent university's engineering department; Pu is a much harder beast to deal with).

I have not seen any sources that indicate they actually acquired any fissile materials.