In reply to Elsier:
There's a few areas which are within walking distance from a train station, with more than enough climbing for a week. From Fontainebleau Avon: Calvaire, Mont Ussy, Roche Hercule, Rocher d'Avon, Mont Aigu; from Nemours: Petit bois, Rocher Gréau, Mont d'Olivet & Mammouths a bit further; from Montigny: Restant du long rocher and Marion des roches; from Bois-le-Roi: Rocher Canon. If staying in Paris (but why?) you can also take the RER D to Malesherbes and walk to Buthiers from there.
If you need a pad delivered to you it seems this company will deliver it to a train station: https://fontainebleau-crashpads.com/#
Getting mats on the train can be a pain if you travel during commuting hours in the same direction as commuters (i.e. towards Paris in the morning and from Paris in the evening), otherwise I'd say it's fine. Parisians do it every weekend. Maybe half of train passengers get on/off the train at Melun.
Renting a bike opens up some more possibilities, from Fontainebleau town the Fontainebleau-Barbizon "trail" will get you safely to Apremont for instance, and even to Cuvier from there (you do not want to cycle on D607, or D152; not a fan of D409 either, and even Route Ronde (D301) can be sketchy during commuting hours and on weekends).