Of course you will be limited, business is a broad area. If your major was in say civil engineering you wouldn't ask if you might not be able to get into another branch e.g. mechanical engineering. If you want to do something very specific and technical then you need the "right" degree, otherwise it's not a problem however.
IMO marketing and finance/accountancy are very good backgrounds, both in breadth (they are vital in all businesses) and scope - the further up the ladder you go the more you'll find people come from either marketing or accounting/finance backgrounds. This is very particular right at the top, for example in a civil engineering firm just about everybody in the board room will be either a marketer, accountant or civil engineer.
If you don't particularly want to go into a 'marketing' job per se, perhaps a slightly broader degree would make it a little easier initially. Once you have relevant work experience however the actual nature of your degree matters less and less. If you're shooting high you'll be looking at getting an MBA eventually anyhow.
In short, just get a good degree and good work experience. A summer job is a big help, once you're done you'll hear a lot of job application replies saying either "lacking experience" or "overqualified".
