I'm in the same position as you -- I've been working in Sweden for the last 5 years, and I can tell you from experience that certain things will be difficult or nigh impossible if you don't have personnummer.
Talking specifically about banking, no, it won't be possible at all. Even with personal number you might run into difficulties if you are only a student and are not otherwise working in the country -- you'll need to convince them that you actually need a Swedish bank account. Obviously, when you are working, you are already required to have a personnummer, and setting up an account would be easy in this case.
Other things that might be difficult or impossible without a personal number:
- Buying a car from a private person
- Signing an accommodation contract
The list goes on. Personally, I don't see why you are surprised by this -- imagine similar situations in your own country and see how that will work out. Besides, for EU citizens it's relatively easy and painless to obtain a personal number.
But you should already have a temporary personal number which is issued with the cooperation of your education institution -- I had it when I first arrived in Sweden and didn't have to do anything special. These numbers have a letter in them as the fourth-to-last digit to distinguish them from ordinary PNs.
At any rate, if you want to obtain a Swedish PN, you need to do only two things:
Register with Migrationsverket, the Swedish migration authorities. Even as EU citizen (which I assume you are), you are required to register your temporary residence if you stay in the country for more than three months. Note that there are similar requirement in other EU countries, so this should not surprise you. It's standard procedure, you should only present your letter of acceptance from the educational institution along with your passport, and it takes a week or two at the most. You should also provide some information about your current accommodation.
Apply also for a Swedish personal number from Skatteverket (the Swedish Tax agency, which among other things, issues personal numbers). This could be tricky if you are in Sweden only for a year or two (you should have a valid cause to request one), but if you are persistent and persuasive, you might be able to get one -- some of the elderly ladies in Skatteverket are charmed easily, especially if you are polite and try to talk to them in Swedish.
Consult the Skatteverket website for some more information, or visit their local office.