Hi @alaina1,
Thank you for your question and welcome to the Bunnings Workshop community, it is fantastic to have you with us.
I'll start by saying any work on an exposed balcony such as this should be done while wearing a Fall Protection Harness. If you are not comfortable with this work, I'd suggest contacting a handyman for their assistance.
Where screwing is not possible, adhesive hooks, double-sided tape and zip ties are going to be your best friend. Unfortunately, you're never going to get as strong an attachment as you would with screws and mechanical fixings into the existing structure, but it will give you a solid solution that won't damage your rental.
I'd suggest using a bird net with enough size to cover the space easily. Something like this Backyard Farmer 4 x 10m x 5mm Anti Bird Net would likely be more than adequate.
Are there any holes in the top rail where the shutters run that can be utilised? If there is, you could use zip ties to attach the netting to the rail.
If not, I'd suggest using Gorilla 1.5m Black Mounting Tape to attach some small timbers such as these 42 x 19mm 1.8m DAR Pine Premium to all sides and of the opening. If there are holes that can be used to mount at the top, you can skip putting this on the top.
You can then screw the netting into the timber using Zenith 8G x 15mm Galvanised Button Head Stitching Timber Screws.
When you plan to leave the rental, you can unscrew the netting and pry the timber off. There may be some adhesive residue, however this is fairly easily removed with something like Goof Off Adhesive Remover.
Allow me to tag @TedBear, @Jewelleryrescue and @Noyade to see if they have any other ideas.
Let me know if you have any further questions.
Jacob
