Mulch, and place "rails" in it for the mower - something hard like concrete pavers or 2x5 lumber (treated) placed far enough apart for the wheels of the mower to roll on.

When I was setting up my vegetable garden, I started with the round bare patch made by my former above-ground pool. We had the pool when the kids were young, they had lots of fun but no one used the pool for two summers so we took it down. Left with a bare circle, I decided to make a veggie garden. Started one spring by having it completely roto-tilled. Then I found large pieces of linoleum a neighbor was throwing away. Laid them out across the diameter of the circle, breaking the garden into two halves. Then placed a few bricks to hold the pieces of linoleum down. I now had access from the outside of the circle and from the center path. This has worked well for me for about ten years.