In South Australia, the Summary Offences Act, 1953 declares that crossbows are "Prohibited Weapons".
In particular (these are extracts from the Act as at 1 April, 2013, so it would pay to check the legislation is still current):
(1) A person who—
(a) manufactures, sells, distributes, supplies or otherwise deals in, prohibited weapons; or
(b) uses or has possession of a prohibited weapon,
is guilty of an offence.
Maximum penalty: $20 000 or imprisonment for 2 years.
(2) It is a defence to prosecution for an offence against subsection (1) to prove that the defendant was, in accordance with—
(a) Schedule 2; or
(b) a declaration made by the Minister under subsection (3),
an exempt person in the circumstances of the alleged offence.
Schedule 2 provides, amongst other things:
"9—Sport and recreation
A person is an exempt person for the purposes of an offence of use or possession of a prohibited weapon under section 21F(1)(b) of this Act if the person uses or has possession of a prohibited weapon in the course of participating in a lawful and recognised form of recreation or sport that reasonably requires the use or possession of the weapon."
In addition, the sale of crossbows is limited to approved manufacturers and retailers (and, once a crossbow is purchased by an individual, that person is prohibited from re-selling it).
The sale, distribution or supply to, or dealing in crossbows with, a person who is under the age of 18 years is prohibited.
Please note Section 21F (14) states "A person (being a person who is otherwise entitled to do so) must not use or have possession of a prohibited weapon unless he or she does so in a safe and secure manner." This implies that crossbows must also be stored securely when not in use.
Archery Clubs in South Australia may allow crossbows to be used at their venue (you will need to check first), but only if you have first undertaken a recognised course of archery instruction and are a bona fide and a fully financial member of that, or another, Club affiliated with Archery Australia.
Other conditions may also apply (such as limits on crossbow poundage). And, unless you are a member of the Club for a particular venue, then the use of crossbows at that venue will be restricted to only when that Club is conducting a programmed shoot.