For a very smart approach, it is good to autoclave all vessels. However, in practice there is no need for this, unless someone next door makes out DNase in preparative quantities, so that it could pulverize around the rooms....
In theory, it's "best practice" to autoclave sterile all glassware before using.
In reality, as long as the containers are washed very well & rinsed in di-water, then they are clean enough for storing DNA-extraction buffers. If you are working on human DNA samples, ancient DNA, etc then there are specific additional steps to use. But if you work in that sort of lab, you'd be getting that training.
Ask your PI what is "standard practice" for the lab you are working in and follow their recommendation.