Whether you are planning an airstrip for your own use, only, or one for your mates and other aircraft types makes a lot of difference and, in Victoria, at least, a planning permit is not required for a private airstrip, only for a public aerodrome. A planning permit is required for any excavation works.
If you follow the guidelines in CAAP 92 you will have a strip like a public aerodrome, far in excess of what is really needed for a Piper Cub. So if your space is limited then you are able to make a strip which is suitable for your Cub but nowhere near meeting the CAAP guidelines.
I have recently been through this, constructing a strip for a similar STOL aircraft. All of my neighbours have been supportive, except one. He has had the police, CASA and the council planning department check me out. Nothing untoward was found, by any of them, but I had been careful to obtain all necessary permits (only for the hangar, there was no excavation required).
There is a previous thread on this subject
Legal Question - Flying From Private Property?