Finance is an extremely broad field.. any idea what business you want to focus on? trading? commercial? m&a? market risk? Undergraduate majors don't necessarily matter but a previous internship in finance goes a long way in making you stand out. Do you have any banking connections? If you are able to get your foot into the door to interview and impress them with your enthusiasm and knowledge of finance, you might have a decent shot.
Without classes like corporate finance, econometrics etc, and no experience you'll have to really demonstrate you know what you're talking about. Having some knowledge of stocks, company valuations and such wouldn't hurt. Disclaimer: if your genuine interest to get into finance amounts to "i want to make tons of $$", you might want to rethink lol.
If I was to be realistic though, I would say its a pretty slim chance right now given the current market and competition for so few spots. you'll never know if you don't try though so good luck! just have a plan b