Oh, what a fascinating question! I think it is still possible, but maybe not in the old way. For instance, there was this guy, the Yak man, who has gone all over the place by bicycle. I think that was a true adventure.
There might also be inspiration in the stories of Richard Halliburton (The Royal Road to Romance and other writings), even though they are very dated now. They certainly fired my imagination when I was a kid. If I had had the means when I was young enough to do it, I would have loved to follow in his footsteps.
I think certain professions are an adventure, either by their own nature or by where they take you, but maybe that is not what you mean.
Another fabulous adventure would probably be to follow the Silk Road all the way from one end to the other. There are still strange things and wonders to be found. The days of Marco Polo and Prester John will never return, but if you follow the threads of research into thousands of subjects, they lead into worlds that are still magical and amazing.