By plane
It's possible to get into Barkly by air. There's plenty of land available, which leads to plenty of places for small airplanes or helicopters to land. The roadhouse Barkly Homestead has 1-2 kilometres long gravel airstrip available.
By bus
Greyhound Australia (http://www.greyhound.com.au/) tel. 1300 473 946, Australian phones only stops on most places of the Stuart Highway. Every day the bus from Alice Springs leaves at 7:30PM and arrives in Tennant Creek about 6,5 hours later. From Darwin, the coach leaves exactly at noon, passes Katherine at 5:25PM and arrives at Tennant Creek at 2AM 14 hours later. Bus travel is expensive, from Alice Springs to Tennant Creek costs AU$169, while from Darwin to Tennant Creek costs around AU$242.
Bus travel can be tedious though - sure, it's a way to get in, but getting around these vast distances is a nightmare. Having your own transport is definitely a better option.
By car
Although Barkly Tableland is one of the remotest places in Australia, it is quite easy to get in with a 2-wheel-drive-car using sealed roads. The Stuart Highway road 87 is a sealed road that crosses the region from north to south, and your best bet if coming in from Darwin, Katherine or Alice Springs. Incoming traffic from Queensland goes via the Barkly Highway road 66, another sealed road in perfect condition. Even while using these quality roads, you are still in the middle of the Outback, so get plenty of fuel, drinks and food on board and make sure your car is in top condition.
More adventurous options are available using unsealed roads, but don't even think about these without having your car in top condition. Some of these roads can be done with a 2WD-car, but it's not recommended as they can be rough. Better rent a 4WD-car. Also make sure you carry plenty of fuel, drinks and food, as you might not see anyone here in days or even weeks. When in danger, never leave your vehicle as then you are a lot harder to be spotted from the air by rescue services.