What is the min. amount of money would it take to travel and backpack Europe for a summer?
Depends on where you’re travelling, how many countries/cities and how budget you’re willing to go.
The cheapest accommodation is camping or couchsurfing.
http://couchsurfing.com
Otherwise, you need to budget €20-25/night for a hostel dorm bed.
If you buy all your food in a grocery store (for example, eat sandwiches and fruit, or make meals in the hostel), you can travel on €10/day. If you eat street food and in cheap restaurants, you need about €30/day.
Budget airlines are sometimes the cheapest way to travel between cities. Ryanair occasionally has deals for free or nearly free tickets. The downside is that the airports they use tend to be a bit outside a major city, so you often have to take a shuttle bus for an hour or more. Price depends on city, but it often €5-20.. still a good deal though.
http://ryanair.com
Trains are another option, and if you take an overnight one you don’t have to pay for a hostel that night. I think Eurail passes are only worthwhile if you are travelling a lot, especially since you pay more for seat reservations, overnight trains and many other extras.
http://bahn.de
I can travel easily on €50/day, but I’m also willing to live very cheaply during a trip. €100/day would be more comfortable.