Luggage storage at Naples train station/Herculaneum

You can store your luggage at the Naples train station. They have a room for that. Watch out for the touts who will offer to help you find the luggage storage at the Naples train station and who want a tip in return. Then take the local public transport to the archeological museum which is one stop away from the train station as I recall. Depending on how much time you have you could also delve a little into Naples itself. We took the Circumvesuviana train to Sorrento and then the bus to Positano. I'm not sure if the train goes past Herculaneum. The train is way less expensive than 40 euros/hour and not difficult.

Luggage storage at Napoli Centrale is a door or two away from the police station, near the side exit of the station. I leave luggage there often.

Well yi, you don't indicate what season you are planning to travel (which dictates when it would be getting dark), or what time you would be arriving in Naples by train. Trying to do everything you list in one day by public transportation would involve lots of transport connections while lugging your baggage around. For three of you, taxis can sometimes make more economic sense than when just one or two people are involved. You need to balance time, hassle and cost. How about this idea:
1. Arrive Naples, store your bags at the train station;
2. Go to the Archeological Museum, either by public transport or taxi, visit the museum and a little more of Naples.
3. Return to the Circumvesuviana section of the train station, take that train to Herculaneum which is just a few minutes, maybe 20 or 25, without your baggage. It is about a 10 minute walk to the ruins, each way. Visit the ruins, then return to take that short train ride back to Naples.
4. Pick up your bags and then get a taxi or car service directly to your door in Positano, which should probably run about €100 or a little more.

There is no bag storage at the Herculaneum train station. The train runs every half hour or every hour, depending on the time of day and the time of year. While there's a schedule, it's not very reliable - the trains are often late. There may be storage at the sight, but it's a 10-minute downhill walk from the station, which means a somewhat longer uphill walk at the end of your visit. For me, it would make sense to leave luggage in Naples to visit the museum. But I would probably pay for a driver to wait with the luggage rather than waste the time going back and forth from Naples by train.

There are precise instructions in the RS books for getting from the Naples train station to the museum. It's a short metro ride and the museum is right above the stop, but it's a bit of a walk from the trains to the metro.

Napoli Centrale train station has a very nice room for luggage storage. Last year the storage cost was 6 Euro per bag, and I had to present my Passport to the Attendant. The room is at the end of a long quiet hallway. Ask one of the many cops in the main area how to find the room.

I love the Napoli Centrale train station. Huge waiting area with seating arrangements everywhere. They have a huge deli/cafeteria (McDonald's is on the other side) that I think makes the best caffe' ever.