Oakleigh is a small suburb and the visitor-facing part of it — the Eaton Mall precinct and the streets immediately surrounding it — is entirely walkable. From Oakleigh Station to Eaton Mall is about five minutes on foot along Portman Street. The pastry shops, cafes, souvlaki restaurants, and delis worth visiting are all within a few hundred metres of each other.
There is no need for a car once you are in Oakleigh if your focus is the dining precinct. If you are combining Oakleigh with a visit to Caulfield Racecourse, the simplest approach is to take the train to Caulfield Station (on the Frankston line) for the racecourse and then continue by train or taxi to Oakleigh for food.
Cycling is possible — the area is relatively flat — but there is no dedicated cycling infrastructure around Eaton Mall itself.