If things continue going as they were for the last years: government printing money and people trying to move savings anywhere but in the currencies, then yes, all printed currencies will become worthless and if people run to Bitcoin, it will be impossible to have "budget deficits" and very hard to tax many people.

Without government people would have to find ways to voluntarily pay for things they like to have. E.g. roads, insurance etc. Some roads will be paid, some roads will be sponsored by merchants that do business on the streets, some roads will be owned by big malls etc. The market is different from the state not in that it does something differently, but that there are millions of ways to satisfy demand instead of just one.

Read more on possibilities in "Practical Anarchy": http://freedomainradio.com/FreeBooks.aspx