Train based travel in France

A lot depends on where in France you want to go, what you want to see, and how long you have to travel. There is good train connection to almost all cities and train or bus transportation from the city to smaller towns/villages but some areas are not as well served as others and it makes sense to rent a car for them if possible - Provence and Brittany come to mind, and Normandy to a lesser extent. I traveled around France for a month using a combination of trains and rental cars , I used cars for the villages of Provence, Normandy, and Brittany. I used train for Riviera, part of Provence, Alsace, Rhone-Alpes regions. If staying in Paris for any length of time, there are innumerable day trips to be done by train.

Your question: " Is it feasible to travel outside of Paris without a car?"

Yes, to those destinations served by rail.

My experience has been the same as Nancy's.

As in most countries, the train system does not have rail lines or regular service into certain rural areas.
The first planning task is to determine which of your destinations are not served by rail.
Then, either delete those destinations from your itinerary or decide that you will rent a car to see those destinations on your itinerary that are not served by rail.

Here's a discussion by Rick of the subject:
http://www.ricksteves.com/travel-tips/transportation/trains/france-rail-passes

The French trains are generally pretty great, but they only go where they go. You will also note that Paris is the great rail hub of France and that it will often be faster to travel through Paris than in a straight line. You should plan your itinerary accordingly.

I love the trains but still find that wheels of ones own greatly enhance the experience in some places. One strategy is to take the trains from region to region and rent a car locally to explore and see a bit of the countryside.

You pretty much can't do much with the Dordogne, Provence or Burgundy without your own wheels.

Yes. French public transport is generally very good within and between large cities, but in rural areas both rail and bus services can be very limited, and infrequent when they do exist. Country bus services in particular are often timed to take schoolchildren and shoppers from villages to the nearest larger town in the morning and back again in the afternoon, not for people who want to make day trips to the villages.

Note that if you must change trains in Paris, you may also have to change stations, so allow time for that. In 2013 I spent a month in France using public transportation. A car would have been more convenient for some smaller towns, but the train/bus combination worked well for me.

As you read about each destination in the book, the start and end of each chapter have the best summaries of transportation issues by train, bus, or car for that town. In the intro on pages 10 and 12, you can see that the route recommendations are different depending whether you prefer all car or all public transport, but a mix of the two is also common. If you want a car for more than one region, then it often makes sense to keep the car in between those regions, as well, especially if you're getting a better price for a weekly or longer rental.