In brief: Yes, you can store your non-hazardous chemicals with your hazardous chemicals provided that incompatible chemicals are segregated. The Australian Standard AS/NZS 2243.10.2004 gives guidance for the storage of chemicals. It states:
Other chemicals that are neither dangerous goods nor hazardous substances (e.g. starch or distilled water) may also be kept in the store.1
Chemicals should be stored segregated on the basis of their physico-chemical attributes. The Dangerous Goods Code is a good foundation for segregation of incompatible chemicals, bearing in mind that there can be incompatibilities within a Dangerous Goods Class, particularly DG class 8, and where a chemical is classified with more than one DG Class. Generally, but not always chemicals classified as Dangerous Goods are often classified as Hazardous.
The term hazardous generally refers to an adverse effect on human health; for more information on this see http://www.safeworkaustralia.gov.au/sites/swa/whs-information/hazardous-chemicals/dangerous-goods/pages/hazardous-substances. Whilst chemicals previously classified as hazardous substances and Dangerous Goods now come under the GHS for workplaces, the Dangerous Goods Code is still a requirement for road transport and a good basis for segregation of incompatible chemicals in storage.
We previously answered a similar question, which has some links to some useful information for the school setting, see Chemical Safety. We will be providing some information on chemical storage guidelines in the Chemical Handbook which we are developing.
1 Standards Australia. 2004. AS/NZS 2243 Safety in Laboratories, Part 10: 2004 Storage of chemicals. Sydney, Australia. (p23) Reproduced with permission from SAI Global Ltd under Licence 1407-c117
Science ASSIST, List of recommended chemicals for science in Australian schools, https://assist.asta.edu.au/resource/3052/list-recommended-chemicals-science-australian-schools (March 2016)
‘Chemical management’, Victorian Department of Education and Training website, http://www.education.vic.gov.au/school/teachers/management/Pages/chemica... (Updated June 2017)
‘Chemicals and hazardous substances’, Qld Department of Education website, Click on the 'Chemicals and Hazardous Substance' link on the following web page, https://education.qld.gov.au/initiatives-and-strategies/health-and-wellbeing/workplaces/safety/hazards (Link updated May 2019)
'Chemical storage and handling for minor quantities in laboratories' University of Melbourne Website, https://safety.unimelb.edu.au/__data/assets/pdf_file/0009/1715346/chemical-storage-and-handling-for-minor-quantities-in-laboratories.pdf (November 2015)
'Chemical-Storage-Guidelines' Charles Sturt University website, https://web.archive.org/web/20170517152745/http://www.csu.edu.au:80/__da... (March 2018: Link to original resource broken and replaced with a copy archived on the Internet Archive's Wayback Machine).
'HS404 Dangerous Goods Storage Guideline' University of NSW website, https://www.gs.unsw.edu.au/policy/documents/HS404.pdf (August 2016)
'Storage of Chemicals', Australian National University website, https://www.google.com.au/url?sa=t&rct=j&q=&esrc=s&source=web&cd=10&cad=... (June 2011)
Storage of minor quantities of hazardous substance in laboratories, University of Wollongong school of chemistry website, https://smah.uow.edu.au/content/groups/public/@web/@sci/@chem/documents/doc/uow043864.pdf (Sept 2009)