Barbells alone are more versatile than first impressions would reveal. Olympic lifts are comparable enough to back squats and deadlifts that a squat rack is technically unnecessary if you actually perform the snatches and cleans with enough frequency and heavy weight. (High volume is not generally necessary though.) There is no doubt that Olympic lifts alone can develop the primary squatting and pulling muscles involved to a level of excellence suitable for all but those with niche requirements.
Olympic lifts + various barbell lifts from the ground can cover the legs and back, but the upper body is a little neglected still, so throw in "landmine" movements by sticking one end of the barbell in a corner and now you have a sort of makeshift dumbbell that acts like a lever. More exercises are available for the upper body then, and even the chest can become involved.
If I were to make this a short list because a barbell alone is not satisfactory, the minimal two items to add would be Olympic rings and a dip belt—or alternatively a "suspension trainer" instead of Olympic rings because suspension trainers often have footholds to provide additional leg exercises. Dips, pull ups, suspended chest flyes, suspended pushups, and suspended rows then become available. The dip belt will allow adding resistance to these exercises with the barbell plates already on hand. A bonus feature I personally noticed and employed is some dip belts make for a good attachment to enable T-bar rows in the landmine position.
In summary a well-rounded, very minimal gym suitable for hobbyists could include just three things:
- a barbell
- Olympic rings (or a suspension trainer)
- a dip belt
While a home gym with Olympic rings can ensure the development of the pecs, it cannot usually provide the opportunity to develop the social character and connections offered by being in the right place and right time in a public gathering ground.