If you want to simply 'walk' around Europe the answer is yes, if you hope to try and live of the land as you go, then that would be a little tricky.
I know a man who each year spends at least a month in Sweden and he takes a small folding fishing rod to supplement his food intake but I don't think you'd have much success doing that elsewhere.
Europe is crisscrossed with hundreds of long distance footpaths most serviced by mountain huts, hostels or tourist houses and as most of them connect to each other you could keep walking for as long as your money lasted.
Here are a few
France has an extensive system of interconnecting footpaths, some good for a days hike while others will take weeks to complete. Wild camping is not permitted, however if you are sensible and far from civilization you will get away with a 'stealth' camp.
A list of French routes may be found here.
The GR65 would take you from central France to the Atlantic coast of Spain, that will keep you busy for a month or two.
Germany offers the excellent 'Westweg' (Black Forest Trail), links
here and
here.
Hungary has several well supported and well marked routes, the 'Kéktúra' or 'Blue Tour' will take you from the Austrian border to the north east of the country, maps, photographs and English text may be found
on this website.
Romania provides a slightly more ruffty-tuffty walking experience in the Fogaras Mountains of Erdély (Transylvania).
This English language website will provide you with photographs, routes etc.
Turkey will give you a real sense of adventure on its long distance paths, the 'Lycian Way' provides a varied landscape and many interesting encounters with man and beast throughout the thirty odd days it will take you to complete. An English language website may be found
here.