Not under tenancy law.
Even if your tenancy agreement says that you have to take out contents insurance, this is inconsistent with the Residential Tenancies Act so you don’t have to comply with it. Clauses which say that you agree to indemnify the landlord against any loss you may suffer are also invalid. For more information on invalid clauses, see Extra terms and standard leases.
A tenant is entitled to compensation from the landlord if the landlord breaches their obligations and this causes the tenant a loss (section 83(d) of the Residential Tenancies Act). The landlord’s obligations in relation to the premises and doing repairs are in clauses 54-62 of the Standard Lease.
If the premises were not provided or maintained reasonably secure, and this allowed somebody to break in and steal or damage your property, you would suffer a loss that was caused by the landlord’s breach.
Similarly, if the property leaks water and your goods become damaged, if the landlord was in breach by failing to provide or maintain the premises in a reasonable state of repair (ie waterproof) then this would be a loss caused by the landlord’s breach.
If the house burns down as a result of an electrical fault that the landlord could have avoided by carrying out repairs or reasonable maintenance, again, your landlord’s breach has caused a loss.
In all these situations, your landlord could be ordered to pay the cost of repairing or replacing your goods. Compensation can also cover non-economic loss that you couldn’t have avoided by taking reasonable steps. So compensation may cover goods that cannot be replaced, like photos or letters. As long as it was reasonably predictable that the landlord’s breach could result in this kind of loss or damage, and the loss results from a breach, you are entitled to compensation.
Breaches by your landlord can also impact on your health, safety and enjoyment of the premises. Your landlord can be ordered to pay compensation for these types of losses as well. If your landlord wants to avoid this significant liability, they can take out landlord’s insurance. If you are trying to persuade your landlord to take out insurance, or to take their obligations under tenancy law seriously, you may find this High Court case useful: Northern Sandblasting Pty Ltd v Harris  HCA 39.
However, if damage occurs as a result of your breach of the tenancy agreement, then you are not entitled to compensation for your losses. If you fail to notify of a repair issue which the landlord had no reasonable way of knowing about, and this problem damages your goods, you won’t get compensation for them.
You also have to take reasonable steps to reduce your loss. So, if you knowingly leave your computer under a roof leak, you will not receive compensation for damage.
However, tenants may consider it worth paying for contents insurance where it may be difficult to prove a breach by the landlord, or even just to save the hassle of claiming compensation when the landlord is in breach.
It is always a good idea to have contents insurance to protect your belongings. If you claim on your insurance because of your landlord’s breach, you can seek the excess from your landlord.