Beware of trusting any financial news services or networks - or financial news from mainstream sources. They are usually full of shit, and by the time you learn of something, it's almost always too late to do anything about it.
As to the bank thing, yeah, it's actually not too big of a deal in and of itself. The problem is that markets are a confidence game (not in the conman sense, but in the sense of relying on trust to work). So, you don't actually need people to even think it's a terrible thing for it to put stress on the system. All you need is for people to think that other people might think Cyprus to be a terrible thing.
There are two major problems with western capitalism at the moment. The first is that, in the financial sector, what is going on isn't capitalism. It is best described as an incredibly intricate social networking game where money is the score for who is winning. At this point, what goes on in Wall Street, London, Tokyo, and etc. has virtually nothing to do with how many tractors Caterpillar made last year, or if Airbus is selling a lot of jets this year. The companies (a term best used loosely when applied to Wall Street) that are involved with financial markets as a primary business have essentially become giant wallet-filling machines by creating an enormous amount of opaque transactions that they skim a little off of for the service of creating them.
I could really go into that, and will if anyone invites me to, but in general, what we have are some entities whose entire focus and motivation is to create an incredibly complex and hard to understand tangle of nearly infinite transactions, expanded every day. These systems tend to be, uh, vulnerable to a lack of confidence, since no one person can possibly comprehend even a sector of it. Oh, and one of the primary things being traded around? Debt. Government debt, city debt, company debt, mortgage debt, credit card debt...
I'm fairly libertarian, so this isn't coming from a leftist, but these people, in my opinion, are a menace.
The other big problem we have is that governments in western countries spend like drunken sailors with poor impulse control. In Keynsian terms, we should be enacting government spending programs during the bust years, and saving during booms. Instead, what we have been doing is spending during the boom years, and spending more during the busts. Whereas a reasonable keynsian approach would even out the pain and pleasure, instead we pump up booms, which increases the inevitable bubbles, and we use up all our money.
Although it is true that a government can run a permanent debt without any problems - and possibly should - what a government cannot do is permanently increase its debt. At some point, the interest on the bonds it sells will be more than it takes in by taxation and state-run enterprises (which tend to suck as revenue generators anyway, though not always), in which case you are fucked.
In this case, suppose Cyprus rejects the bailout thanks to popular outrage? Well, it might go something like this:
Cyprus's credit rating is downgraded, to junk, leaving many entities (like, say, hedge funds or other institutions) with worthless bonds. They have to report these losses to their own investors. These investors get scared, and start pulling money out. This loss of capital means that the funds (or whoever) need to sell other bonds (say, of other European countries) to raise capital, since their loss of investors means that the banks are less willing to lend to them. Some of these entities are able to sell their bonds, while others fail, but the mass sale drops the confidence in those bonds that were sold. Those bonds were naturally the weakest the funds thought they could get rid of, and so those weaker countries now have trouble selling more bonds to prop up their debt. Soon, one of them can no longer afford to offer the higher interest payments now necessary to entice investors to take on the risk of a default. It defaults in turn, and wheeeeeee, here we go, as the cycle picks up speed. And I haven't even mentioned all the really fun stuff like collateral calls, credit default swaps, and so forth, that basically tie everyone together in this giant game of lemmings.
The really fun fact is that all of the above can happen even if Cyprus doesn't actually default, but people think they will.