Yes - Forest Hill School has a sports hall that's available in evenings and at weekends to public. St Dunstan's also has a gym and swimming pool but when I enquired in the summer it was not at that time available to the public although it used to be.
Pity we have to wait so long for the pool to re-open with decent facilities. Only other option is The Bridge but presumably that's not within walking distance.
Thee is a women only gym on the station approach at Sydenham but have not tried it - at least only one station away. However you have to make an appointment to go there so not really ideal.