Apart from running the LeLingo bilingual helpline I am also a registered farmer here in France since 2002. Helen Franklin's answer is the most complete so far with one notable exception. As well as being a registered farmer with the MSA (Mutuelle Sociale Agricole) which you must be to exploit your land as a business, you will also have to be registered with your local Chambre d'Agriculture. The land you wish to cultivate must also have been declared for EU subsidies BEFORE the end of 1992 otherwise it will not be eligible for subsidies even if you become a registered "Exploitation Agricole" or "Chef d'Entreprise Agricole". A further complication will be that in your department there will be a minimum surface of "terres"* under your ownership or control for you to be able to register - even if you qualify in all other areas. I know this because I have been through the "hoops" in the Lot for my own 23 hectares and have helped other people register.
A further complication is that to become a registered farmer in France you will need to prove you have the qualifications for it before you can be accepted and registered, rather like the Chambre des Métiers where an Artisan has to prove relevant qualifications OR experience before he or she can register. This is why there is so much opposition in France to the Auto Entrepreneur status who do not have to prove their bona-fides to be able to work and pay a reduced tax and social charge rate for, now at least, the first three years.
Finally, beware the trap of renting your land to a neighbouring farmer so that he exploits it 'for you' and you get a small income from it. You may/will find that you cannot get him off the land and, if you decide to sell, depending on your individual circumstances there may be some serious capital gains surprises for you at the point of completion.
In my experience, both in the UK and here in France subsidies are like a double edged sword often taking more in the long run than is given.
I hope that information is useful and wish you good luck.
* Your relevé de parcelles in you title deeds should indicate what the official designation of each part of your land is. If they are not "terres" then they are no good for arable or cultivation anyway.